diff --git a/docs/plot/plotdefs.json b/docs/plot/plotdefs.json
index a6d70273e1..f55f5e0ab0 100644
--- a/docs/plot/plotdefs.json
+++ b/docs/plot/plotdefs.json
@@ -1057,7 +1057,7 @@
         "lonmax": 180,
         "lonmin": -180,
         "name": "peirce_q_square",
-        "projstring": "+proj=peirce_q +lon_0=25 +type=square",
+        "projstring": "+proj=peirce_q +lon_0=25 +shape=square",
         "res": "low",
         "type": "poly"
     },
@@ -1068,7 +1068,7 @@
         "lonmax": 180,
         "lonmin": -180,
         "name": "peirce_q",
-        "projstring": "+proj=peirce_q +lon_0=25 +type=diamond",
+        "projstring": "+proj=peirce_q +lon_0=25 +shape=diamond",
         "res": "low",
         "type": "poly"
     },
@@ -1079,7 +1079,7 @@
         "lonmax": 180,
         "lonmin": -180,
         "name": "peirce_q_horizontal",
-        "projstring": "+proj=peirce_q +lon_0=25 +type=horizontal",
+        "projstring": "+proj=peirce_q +lon_0=25 +shape=horizontal",
         "res": "low",
         "type": "line"
     },
@@ -1090,7 +1090,7 @@
         "lonmax": 180,
         "lonmin": -180,
         "name": "grieger_triptychial",
-        "projstring": "+proj=pipeline +step +proj=ob_tran +o_proj=peirce_q +o_lat_p=-45 +o_lon_p=45 +type=horizontal +scrollx=-0.25 +step +proj=affine +s11=-1 +s12=0 +s21=0 +s22=-1",
+        "projstring": "+proj=pipeline +step +proj=ob_tran +o_proj=peirce_q +o_lat_p=-45 +o_lon_p=45 +shape=horizontal +scrollx=-0.25 +step +proj=affine +s11=-1 +s12=0 +s21=0 +s22=-1",
         "res": "low",
         "type": "poly"
     },
diff --git a/docs/source/operations/projections/peirce_q.rst b/docs/source/operations/projections/peirce_q.rst
index cf97293280..57504032af 100644
--- a/docs/source/operations/projections/peirce_q.rst
+++ b/docs/source/operations/projections/peirce_q.rst
@@ -14,10 +14,10 @@ defaults to a central meridian of 0, it is more common to use a central
 meridian of around 25 to optimise the distortions. Peirce's original
 published map from 1879 used a central meridian of approx -70.
 The diamond and square versions can be produced using the
-``+type=diamond`` and ``+type=square`` options respectively.
+``+shape=diamond`` and ``+shape=square`` options respectively.
 This implementation includes an alternative lateral projection
-which places hemispheres side-by-side (``+type=horizontal`` or
-``+type=vertical``). Combined with a general oblique transformation,
+which places hemispheres side-by-side (``+shape=horizontal`` or
+``+shape=vertical``). Combined with a general oblique transformation,
 this can be used to produced a Grieger Triptychial projection
 (see example below).
 
@@ -43,21 +43,21 @@ this can be used to produced a Grieger Triptychial projection
    :align: center
    :alt:   Peirce Quincuncial (Square)
 
-   proj-string: ``+proj=peirce_q +lon_0=25 +type=square``
+   proj-string: ``+proj=peirce_q +lon_0=25 +shape=square``
 
 .. figure:: ./images/peirce_q_diamond.png
    :width: 500 px
    :align: center
    :alt:   Peirce Quincuncial (Diamond)
 
-   proj-string: ``+proj=peirce_q +lon_0=25 +type=diamond``
+   proj-string: ``+proj=peirce_q +lon_0=25 +shape=diamond``
 
 .. figure:: ./images/peirce_q_horizontal.png
    :width: 500 px
    :align: center
    :alt:   Peirce Quincuncial (Horizontal)
 
-   proj-string: ``+proj=peirce_q +lon_0=25 +type=horizontal``
+   proj-string: ``+proj=peirce_q +lon_0=25 +shape=horizontal``
 
 .. figure:: ./images/grieger_triptychial.png
    :width: 500 px
@@ -73,16 +73,20 @@ Parameters
 
 .. include:: ../options/lon_0.rst
 
-.. option:: +type=square/diamond/horizontal/vertical/nhemisphere/shemisphere
+.. option:: +shape=square/diamond/horizontal/vertical/nhemisphere/shemisphere
 
-    .. versionadded:: 8.2.1
+    .. versionadded:: 9.0
 
     *Defaults to diamond.*
 
-    Indicates the type of transformation applied to the southern hemisphere:
+    .. warning:: This option was wrongly introduced introduced in 8.2.1 with the
+                 ``type`` name, which was inappropriate as it conflicted with
+                 the ``+type=crs`` general hint.
+
+    Indicates the shape of transformation applied to the southern hemisphere:
     ``square`` and ``diamond`` represent the traditional quincuncial form suggested
     by Peirce with the southern hemisphere divided into 4 triangles and reflected
-    outward from the northern hemisphere. The ``square`` type is rotated by 45
+    outward from the northern hemisphere. The ``square`` shape is rotated by 45
     degrees to produce the conventional square presentation. The origin lies at
     the centre of the square or diamond.
 
@@ -96,14 +100,14 @@ Parameters
 
 .. option:: +scrollx=<value>
 
-    For ``horizontal`` type allows a scalar circular scroll of resulting x coordinates
+    For ``horizontal`` shape allows a scalar circular scroll of resulting x coordinates
     to shift sections of the projection to the other horizontal side of the map.
 
     *Defaults to 0.0. Must be a scale between -1.0 and 1.0.*
 
 .. option:: +scrolly=<value>
 
-    For ``vertical`` type allows a scalar circular scroll of resulting y coordinates
+    For ``vertical`` shape allows a scalar circular scroll of resulting y coordinates
     to shift sections of the projection to the other vertical side of the map.
 
     *Defaults to 0.0. Must be a scale between -1.0 and 1.0.*
diff --git a/scripts/build_esri_projection_mapping.py b/scripts/build_esri_projection_mapping.py
index 30f4ad6c66..32920a599b 100644
--- a/scripts/build_esri_projection_mapping.py
+++ b/scripts/build_esri_projection_mapping.py
@@ -710,6 +710,24 @@
         - False_Northing: EPSG_NAME_PARAMETER_FALSE_NORTHING
         - Central_Meridian: EPSG_NAME_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN
 
+- Peirce_Quincuncial:
+    -   WKT2_name: PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_SQUARE
+        Params:
+            - False_Easting: EPSG_NAME_PARAMETER_FALSE_EASTING
+            - False_Northing: EPSG_NAME_PARAMETER_FALSE_NORTHING
+            - Central_Meridian: EPSG_NAME_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN
+            - Scale_Factor: EPSG_NAME_PARAMETER_SCALE_FACTOR_AT_NATURAL_ORIGIN
+            - Latitude_Of_Origin: EPSG_NAME_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN
+            - Option: 0.0
+
+    -   WKT2_name: PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_DIAMOND
+        Params:
+            - False_Easting: EPSG_NAME_PARAMETER_FALSE_EASTING
+            - False_Northing: EPSG_NAME_PARAMETER_FALSE_NORTHING
+            - Central_Meridian: EPSG_NAME_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN
+            - Scale_Factor: EPSG_NAME_PARAMETER_SCALE_FACTOR_AT_NATURAL_ORIGIN
+            - Latitude_Of_Origin: EPSG_NAME_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN
+            - Option: 1.0
 
 # Missing/unclear mappings
 
@@ -748,7 +766,6 @@
 # Polar_Stereographic_Variant_C
 # Quartic_Authalic_Ellipsoidal
 # Adams_Square_II
-# Peirce_Quincuncial
 
 """
 
diff --git a/src/iso19111/io.cpp b/src/iso19111/io.cpp
index ca5a3d977b..fc92dbbdfe 100644
--- a/src/iso19111/io.cpp
+++ b/src/iso19111/io.cpp
@@ -10209,6 +10209,8 @@ PROJStringParser::Private::buildProjectedCRS(int iStep,
                 }
             } else if (param->unit_type == UnitOfMeasure::Type::SCALE) {
                 value = 1;
+            } else if (step.name == "peirce_q" && proj_name == "lat_0") {
+                value = 90;
             }
 
             PropertyMap propertiesParameter;
diff --git a/src/iso19111/operation/conversion.cpp b/src/iso19111/operation/conversion.cpp
index 77e703f360..6bccf18281 100644
--- a/src/iso19111/operation/conversion.cpp
+++ b/src/iso19111/operation/conversion.cpp
@@ -3839,6 +3839,30 @@ void Conversion::_exportToPROJString(
         formatter->addParam("x_0", falseEasting);
         formatter->addParam("y_0", falseNorthing);
         bConversionDone = true;
+    } else if (ci_equal(methodName,
+                        PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_SQUARE) ||
+               ci_equal(methodName,
+                        PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_DIAMOND)) {
+        const auto &scaleFactor = parameterValueMeasure(
+            EPSG_CODE_PARAMETER_SCALE_FACTOR_AT_NATURAL_ORIGIN);
+        if (scaleFactor.unit().type() != common::UnitOfMeasure::Type::UNKNOWN &&
+            std::fabs(scaleFactor.getSIValue() - 1.0) > 1e-10) {
+            throw io::FormattingException(
+                "Only scale factor = 1 handled for Peirce Quincuncial");
+        }
+        const double latitudeOfOriginDeg =
+            parameterValue(EPSG_CODE_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN)
+                        ->value()
+                        .unit()
+                        .type() != common::UnitOfMeasure::Type::UNKNOWN
+                ? parameterValueNumeric(
+                      EPSG_CODE_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN,
+                      common::UnitOfMeasure::DEGREE)
+                : 90.0;
+        if (std::fabs(latitudeOfOriginDeg - 90.0) > 1e-10) {
+            throw io::FormattingException("Only latitude of natural origin = "
+                                          "90 handled for Peirce Quincuncial");
+        }
     } else if (formatter->convention() ==
                    io::PROJStringFormatter::Convention::PROJ_5 &&
                isZUnitConversion) {
diff --git a/src/iso19111/operation/esriparammappings.cpp b/src/iso19111/operation/esriparammappings.cpp
index 8021b95fff..a6f6c6a120 100644
--- a/src/iso19111/operation/esriparammappings.cpp
+++ b/src/iso19111/operation/esriparammappings.cpp
@@ -947,6 +947,33 @@ static const ESRIParamMapping paramsESRI_Equal_Earth[] = {
      EPSG_CODE_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN, "0.0", false},
     {nullptr, nullptr, 0, "0.0", false}};
 
+static const ESRIParamMapping paramsESRI_Peirce_Quincuncial_alt1[] = {
+    {"False_Easting", EPSG_NAME_PARAMETER_FALSE_EASTING,
+     EPSG_CODE_PARAMETER_FALSE_EASTING, "0.0", false},
+    {"False_Northing", EPSG_NAME_PARAMETER_FALSE_NORTHING,
+     EPSG_CODE_PARAMETER_FALSE_NORTHING, "0.0", false},
+    {"Central_Meridian", EPSG_NAME_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN,
+     EPSG_CODE_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN, "0.0", false},
+    {"Scale_Factor", EPSG_NAME_PARAMETER_SCALE_FACTOR_AT_NATURAL_ORIGIN,
+     EPSG_CODE_PARAMETER_SCALE_FACTOR_AT_NATURAL_ORIGIN, "0.0", false},
+    {"Latitude_Of_Origin", EPSG_NAME_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN,
+     EPSG_CODE_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN, "0.0", false},
+    {"Option", nullptr, 0, "0.0", false},
+    {nullptr, nullptr, 0, "0.0", false}};
+static const ESRIParamMapping paramsESRI_Peirce_Quincuncial_alt2[] = {
+    {"False_Easting", EPSG_NAME_PARAMETER_FALSE_EASTING,
+     EPSG_CODE_PARAMETER_FALSE_EASTING, "0.0", false},
+    {"False_Northing", EPSG_NAME_PARAMETER_FALSE_NORTHING,
+     EPSG_CODE_PARAMETER_FALSE_NORTHING, "0.0", false},
+    {"Central_Meridian", EPSG_NAME_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN,
+     EPSG_CODE_PARAMETER_LONGITUDE_OF_NATURAL_ORIGIN, "0.0", false},
+    {"Scale_Factor", EPSG_NAME_PARAMETER_SCALE_FACTOR_AT_NATURAL_ORIGIN,
+     EPSG_CODE_PARAMETER_SCALE_FACTOR_AT_NATURAL_ORIGIN, "0.0", false},
+    {"Latitude_Of_Origin", EPSG_NAME_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN,
+     EPSG_CODE_PARAMETER_LATITUDE_OF_NATURAL_ORIGIN, "0.0", false},
+    {"Option", nullptr, 0, "1.0", false},
+    {nullptr, nullptr, 0, "0.0", false}};
+
 static const ESRIMethodMapping esriMappings[] = {
     {"Plate_Carree", EPSG_NAME_METHOD_EQUIDISTANT_CYLINDRICAL,
      EPSG_CODE_METHOD_EQUIDISTANT_CYLINDRICAL, paramsESRI_Plate_Carree},
@@ -1115,6 +1142,10 @@ static const ESRIMethodMapping esriMappings[] = {
      EPSG_CODE_METHOD_COLOMBIA_URBAN, paramsESRI_IGAC_Plano_Cartesiano},
     {"Equal_Earth", EPSG_NAME_METHOD_EQUAL_EARTH, EPSG_CODE_METHOD_EQUAL_EARTH,
      paramsESRI_Equal_Earth},
+    {"Peirce_Quincuncial", PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_SQUARE, 0,
+     paramsESRI_Peirce_Quincuncial_alt1},
+    {"Peirce_Quincuncial", PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_DIAMOND, 0,
+     paramsESRI_Peirce_Quincuncial_alt2},
 };
 
 // ---------------------------------------------------------------------------
diff --git a/src/iso19111/operation/parammappings.cpp b/src/iso19111/operation/parammappings.cpp
index af1309b3d4..dd94e26706 100644
--- a/src/iso19111/operation/parammappings.cpp
+++ b/src/iso19111/operation/parammappings.cpp
@@ -793,6 +793,12 @@ static const MethodMapping projectionMethodMappings[] = {
     {PROJ_WKT2_NAME_METHOD_ROBINSON, 0, "Robinson", "robin", nullptr,
      paramsLonNatOriginLongitudeCentre},
 
+    {PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_SQUARE, 0, nullptr, "peirce_q",
+     "shape=square", paramsNatOriginScale},
+
+    {PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_DIAMOND, 0, nullptr, "peirce_q",
+     "shape=diamond", paramsNatOriginScale},
+
     {PROJ_WKT2_NAME_METHOD_SINUSOIDAL, 0, "Sinusoidal", "sinu", nullptr,
      paramsLonNatOriginLongitudeCentre},
 
diff --git a/src/proj_constants.h b/src/proj_constants.h
index 815511fe9e..1da6347423 100644
--- a/src/proj_constants.h
+++ b/src/proj_constants.h
@@ -248,6 +248,9 @@
 #define EPSG_CODE_METHOD_COLOMBIA_URBAN 1052
 #define EPSG_NAME_METHOD_COLOMBIA_URBAN "Colombia Urban"
 
+#define PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_SQUARE "Peirce Quincuncial (Square)"
+#define PROJ_WKT2_NAME_METHOD_PEIRCE_QUINCUNCIAL_DIAMOND "Peirce Quincuncial (Diamond)"
+
 /* ------------------------------------------------------------------------ */
 
 /* Projection parameters */
diff --git a/src/projections/adams.cpp b/src/projections/adams.cpp
index 4150fff2ee..b0c2849c4f 100644
--- a/src/projections/adams.cpp
+++ b/src/projections/adams.cpp
@@ -60,7 +60,7 @@ enum projection_type {
     ADAMS_WS2,
 };
 
-enum peirce_type {
+enum peirce_shape {
   PEIRCE_Q_SQUARE,
   PEIRCE_Q_DIAMOND,
   PEIRCE_Q_NHEMISPHERE,
@@ -71,7 +71,7 @@ enum peirce_type {
 
 struct pj_opaque {
     projection_type mode;
-    peirce_type pqtype;
+    peirce_shape pqshape;
     double scrollx = 0.0;
     double scrolly = 0.0;
 };
@@ -143,13 +143,13 @@ static PJ_XY adams_forward(PJ_LP lp, PJ *P) {
         break;
     case PEIRCE_Q: {
             /* lam0 - note that the original Peirce model used a central meridian of around -70deg, but the default within proj is +lon0=0 */
-            if (Q->pqtype == PEIRCE_Q_NHEMISPHERE) {
+            if (Q->pqshape == PEIRCE_Q_NHEMISPHERE) {
               if( lp.phi < -TOL ) {
                 proj_errno_set(P, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
                 return proj_coord_error().xy;
               }
             }
-            if (Q->pqtype == PEIRCE_Q_SHEMISPHERE) {
+            if (Q->pqshape == PEIRCE_Q_SHEMISPHERE) {
               if( lp.phi > -TOL ) {
                 proj_errno_set(P, PROJ_ERR_COORD_TRANSFM_OUTSIDE_PROJECTION_DOMAIN);
                 return proj_coord_error().xy;
@@ -211,7 +211,7 @@ static PJ_XY adams_forward(PJ_LP lp, PJ *P) {
       constexpr double shd = 1.8540746773013719 * 2;
 
       /* For square and diamond Quincuncial projections, spin out southern hemisphere to triangular segments of quincunx (before rotation for square)*/
-      if( Q->pqtype == PEIRCE_Q_SQUARE || ( Q->pqtype == PEIRCE_Q_DIAMOND )) {
+      if( Q->pqshape == PEIRCE_Q_SQUARE || ( Q->pqshape == PEIRCE_Q_DIAMOND )) {
         if (lp.phi < 0.)  { /* fold out segments */
           if (lp.lam < ( -0.75 * M_PI )) xy.y = shd - xy.y; /* top left segment, shift up and reflect y */
           if ( (lp.lam < (-0.25 * M_PI)) && (lp.lam >= ( -0.75 * M_PI ))) xy.x = - shd - xy.x; /* left segment, shift left and reflect x */
@@ -222,20 +222,20 @@ static PJ_XY adams_forward(PJ_LP lp, PJ *P) {
       }
 
       /* For square types rotate xy by 45 deg */
-      if( Q->pqtype == PEIRCE_Q_SQUARE ) {
+      if( Q->pqshape == PEIRCE_Q_SQUARE ) {
             const double temp = xy.x;
             xy.x = RSQRT2 * (xy.x - xy.y);
             xy.y = RSQRT2 * (temp + xy.y);
       }
 
       /* For rectangle Quincuncial projs, spin out southern hemisphere to east (horizontal) or north (vertical) after rotation */
-      if( Q->pqtype == PEIRCE_Q_HORIZONTAL ) {
+      if( Q->pqshape == PEIRCE_Q_HORIZONTAL ) {
         if (lp.phi < 0.)  {
           xy.x = shd - xy.x; /* reflect x to east */
         }
         xy.x = xy.x - (shd / 2); /* shift everything so origin is in middle of two hemispheres */
       }
-      if( Q->pqtype == PEIRCE_Q_VERTICAL ) {
+      if( Q->pqshape == PEIRCE_Q_VERTICAL ) {
         if (lp.phi < 0.)  {
           xy.y = shd - xy.y; /* reflect y to north */
         }
@@ -243,7 +243,7 @@ static PJ_XY adams_forward(PJ_LP lp, PJ *P) {
       }
 
       //if o_scrollx param present, scroll x
-      if (!(Q->scrollx == 0.0) && (Q->pqtype == PEIRCE_Q_HORIZONTAL) ) {
+      if (!(Q->scrollx == 0.0) && (Q->pqshape == PEIRCE_Q_HORIZONTAL) ) {
         double xscale = 2.0;
         double xthresh = shd / 2;
         xy.x = xy.x + (Q->scrollx * (xthresh * 2 * xscale)); /*shift relative to proj width*/
@@ -256,7 +256,7 @@ static PJ_XY adams_forward(PJ_LP lp, PJ *P) {
       }
 
       //if o_scrolly param present, scroll y
-      if (!(Q->scrolly == 0.0) && (Q->pqtype == PEIRCE_Q_VERTICAL)) {
+      if (!(Q->scrolly == 0.0) && (Q->pqshape == PEIRCE_Q_VERTICAL)) {
         double yscale = 2.0;
         double ythresh = shd / 2;
         xy.y = xy.y + (Q->scrolly * (ythresh * 2 * yscale)); /*shift relative to proj height*/
@@ -299,6 +299,91 @@ static PJ_LP adams_inverse(PJ_XY xy, PJ *P)
     return pj_generic_inverse_2d(xy, P, lp);
 }
 
+static PJ_LP peirce_q_square_inverse(PJ_XY xy, PJ *P)
+{
+    /* Heuristics based on trial and repeat process */
+    PJ_LP lp;
+    lp.phi = 0;
+    if( xy.x == 0 && xy.y < 0 )
+    {
+        lp.lam = -M_PI / 4;
+        if( fabs(xy.y) < 2.622057580396 )
+            lp.phi = M_PI / 4;
+    }
+    else if( xy.x > 0 && fabs(xy.y) < 1e-7 )
+        lp.lam = M_PI / 4;
+    else if( xy.x < 0 && fabs(xy.y) < 1e-7 )
+    {
+        lp.lam = -3 * M_PI / 4;
+        lp.phi = M_PI / 2 / 2.622057574224 * xy.x + M_PI / 2;
+    }
+    else if( fabs(xy.x) < 1e-7 && xy.y > 0 )
+        lp.lam = 3 * M_PI / 4;
+    else if( xy.x >= 0 && xy.y <= 0 )
+    {
+        lp.lam = 0;
+        if( xy.x == 0 && xy.y == 0 )
+        {
+            lp.phi = M_PI / 2;
+            return lp;
+        }
+    }
+    else if( xy.x >= 0 && xy.y >= 0 )
+        lp.lam = M_PI / 2;
+    else if( xy.x <= 0 && xy.y >= 0 )
+    {
+        if( fabs(xy.x) < fabs(xy.y) )
+            lp.lam = M_PI * 0.9;
+        else
+            lp.lam = -M_PI * 0.9;
+    }
+    else /* if( xy.x <= 0 && xy.y <= 0 ) */
+        lp.lam = -M_PI / 2;
+    return pj_generic_inverse_2d(xy, P, lp);
+}
+
+static PJ_LP peirce_q_diamond_inverse(PJ_XY xy, PJ *P)
+{
+    /* Heuristics based on a trial and repeat process */
+    PJ_LP lp;
+    lp.phi = 0;
+    if( xy.x >= 0 && xy.y <= 0 )
+    {
+        lp.lam = M_PI / 4;
+        if( xy.x > 0 && xy.y == 0 )
+        {
+            lp.lam = M_PI / 2;
+            lp.phi = 0;
+        }
+        else if( xy.x == 0 && xy.y == 0 )
+        {
+            lp.lam = 0;
+            lp.phi = M_PI / 2;
+            return lp;
+        }
+        else if( xy.x == 0 && xy.y < 0 )
+        {
+            lp.lam = 0;
+            lp.phi = M_PI / 4;
+        }
+    }
+    else if( xy.x >= 0 && xy.y >= 0 )
+        lp.lam = 3 * M_PI / 4;
+    else if( xy.x <= 0 && xy.y >= 0 )
+    {
+        lp.lam = -3 * M_PI / 4;
+    }
+    else /* if( xy.x <= 0 && xy.y <= 0 ) */
+        lp.lam = -M_PI / 4;
+
+    if( fabs(xy.x) > 1.8540746773013719 + 1e-3 ||
+        fabs(xy.y) > 1.8540746773013719 + 1e-3 )
+    {
+        lp.phi = -M_PI / 4;
+    }
+
+    return pj_generic_inverse_2d(xy, P, lp);
+}
 
 static PJ *setup(PJ *P, projection_type mode) {
     struct pj_opaque *Q = static_cast<struct pj_opaque*>(
@@ -316,25 +401,27 @@ static PJ *setup(PJ *P, projection_type mode) {
         P->inv = adams_inverse;
 
     if( mode == PEIRCE_Q) {
-      // Quincuncial projections type options: square, diamond, hemisphere, horizontal (rectangle) or vertical (rectangle)
-      const char* pqtype = pj_param (P->ctx, P->params, "stype").s;
+      // Quincuncial projections shape options: square, diamond, hemisphere, horizontal (rectangle) or vertical (rectangle)
+      const char* pqshape = pj_param (P->ctx, P->params, "sshape").s;
 
-      if (!pqtype) pqtype = "diamond"; /* default if type value not supplied */
+      if (!pqshape) pqshape = "diamond"; /* default if shape value not supplied */
 
-      if (strcmp(pqtype, "square") == 0) {
-        Q->pqtype = PEIRCE_Q_SQUARE;
+      if (strcmp(pqshape, "square") == 0) {
+        Q->pqshape = PEIRCE_Q_SQUARE;
+        P->inv = peirce_q_square_inverse;
       }
-      else if (strcmp(pqtype, "diamond") == 0) {
-        Q->pqtype = PEIRCE_Q_DIAMOND;
+      else if (strcmp(pqshape, "diamond") == 0) {
+        Q->pqshape = PEIRCE_Q_DIAMOND;
+        P->inv = peirce_q_diamond_inverse;
       }
-      else if (strcmp(pqtype, "nhemisphere") == 0) {
-        Q->pqtype = PEIRCE_Q_NHEMISPHERE;
+      else if (strcmp(pqshape, "nhemisphere") == 0) {
+        Q->pqshape = PEIRCE_Q_NHEMISPHERE;
       }
-      else if (strcmp(pqtype, "shemisphere") == 0) {
-        Q->pqtype = PEIRCE_Q_SHEMISPHERE;
+      else if (strcmp(pqshape, "shemisphere") == 0) {
+        Q->pqshape = PEIRCE_Q_SHEMISPHERE;
       }
-      else if (strcmp(pqtype, "horizontal") == 0) {
-        Q->pqtype = PEIRCE_Q_HORIZONTAL;
+      else if (strcmp(pqshape, "horizontal") == 0) {
+        Q->pqshape = PEIRCE_Q_HORIZONTAL;
         if (pj_param(P->ctx, P->params, "tscrollx").i) {
           double scrollx;
           scrollx = pj_param(P->ctx, P->params, "dscrollx").f;
@@ -345,8 +432,8 @@ static PJ *setup(PJ *P, projection_type mode) {
           Q->scrollx = scrollx;
         }
       }
-      else if (strcmp(pqtype, "vertical") == 0) {
-        Q->pqtype = PEIRCE_Q_VERTICAL;
+      else if (strcmp(pqshape, "vertical") == 0) {
+        Q->pqshape = PEIRCE_Q_VERTICAL;
         if (pj_param(P->ctx, P->params, "tscrolly").i) {
           double scrolly;
           scrolly = pj_param(P->ctx, P->params, "dscrolly").f;
@@ -358,7 +445,7 @@ static PJ *setup(PJ *P, projection_type mode) {
         }
       }
       else {
-            proj_log_error (P, _("peirce_q: invalid value for 'type' parameter"));
+            proj_log_error (P, _("peirce_q: invalid value for 'shape' parameter"));
             return pj_default_destructor (P, PROJ_ERR_INVALID_OP_ILLEGAL_ARG_VALUE);
       }
 
diff --git a/test/gie/peirce_q.gie b/test/gie/peirce_q.gie
index 713d2581fd..777ce054bf 100644
--- a/test/gie/peirce_q.gie
+++ b/test/gie/peirce_q.gie
@@ -14,7 +14,7 @@
 ------------------------------------------------------------
 
 ------------------------------------------------------------
-operation  +proj=peirce_q +R=6370997 +type=square
+operation  +proj=peirce_q +R=6370997 +shape=square
 tolerance  10 mm
 ------------------------------------------------------------
 
@@ -138,7 +138,7 @@ accept      170.3972098164   87.6222109978
 expect        -153172.25	215556.52
 
  ------------------------------------------------------------
- operation  +proj=peirce_q +R=6370997 +type=diamond
+ operation  +proj=peirce_q +R=6370997 +shape=diamond
  tolerance  10 mm
  ------------------------------------------------------------
 
@@ -262,7 +262,7 @@ accept       170.3972098164   87.6222109978
 expect        44112.34	260730.62
 
 ------------------------------------------------------------
-operation  +proj=peirce_q +R=6370997 +type=horizontal
+operation  +proj=peirce_q +R=6370997 +shape=horizontal
 tolerance  10 mm
 ------------------------------------------------------------
 
@@ -386,7 +386,7 @@ accept      170.3972098164   87.6222109978
 expect        -11768191.87	260730.62
 
  ------------------------------------------------------------
- operation  +proj=peirce_q +R=6370997 +type=horizontal +scrollx=0.75
+ operation  +proj=peirce_q +R=6370997 +shape=horizontal +scrollx=0.75
  tolerance  10 mm
  ------------------------------------------------------------
 
@@ -510,7 +510,7 @@ accept       170.3972098164   87.6222109978
 expect        -23580496.07	260730.62
 
  ------------------------------------------------------------
- operation  +proj=peirce_q +R=6370997 +type=vertical
+ operation  +proj=peirce_q +R=6370997 +shape=vertical
  tolerance  10 mm
  ------------------------------------------------------------
 
@@ -634,7 +634,7 @@ accept       170.3972098164   87.6222109978
 expect        44112.34	-11551573.59
 
  ------------------------------------------------------------
- operation  +proj=peirce_q +R=6370997 +type=vertical +scrolly=-0.25
+ operation  +proj=peirce_q +R=6370997 +shape=vertical +scrolly=-0.25
  tolerance  10 mm
  ------------------------------------------------------------
 
@@ -758,7 +758,7 @@ accept       170.3972098164   87.6222109978
 expect        44112.34	-23363877.80
 
  ------------------------------------------------------------
- operation  +proj=peirce_q +R=6370997 +type=nhemisphere
+ operation  +proj=peirce_q +R=6370997 +shape=nhemisphere
  tolerance  10 mm
  ------------------------------------------------------------
 
@@ -882,7 +882,7 @@ accept       170.3972098164   87.6222109978
 expect        44112.34	260730.62
 
  ------------------------------------------------------------
- operation  +proj=peirce_q +R=6370997 +type=shemisphere
+ operation  +proj=peirce_q +R=6370997 +shape=shemisphere
  tolerance  10 mm
  ------------------------------------------------------------
 
@@ -1005,4 +1005,330 @@ expect        failure errno coord_transfm_outside_projection_domain
 accept       170.3972098164   87.6222109978
 expect        failure errno coord_transfm_outside_projection_domain
 
+# Test inverse
+------------------------------------------------------------
+operation  +proj=peirce_q +shape=square
+------------------------------------------------------------
+
+#tolerance   1 mm
+# has to bump to this for i386
+tolerance   150 mm
+
+accept      0 90
+expect      0 0
+roundtrip   1
+
+accept      0 0
+expect      8361921.234827487729   -8361921.234827487729
+roundtrip   1
+
+accept      0 -90
+expect      16723842.303160080686   -16723842.303160080686
+#tolerance   2 mm
+roundtrip   1
+#tolerance   1 mm
+
+accept      0 45
+expect      3725360.212758612353   -3725360.212758612353
+roundtrip   1
+
+accept      0 -45
+expect      12998482.090401465073   -12998482.090401465073
+roundtrip   1
+
+accept      45 0
+tolerance   200 mm
+expect      16723842.564696932212   -0.095041956369
+roundtrip   1
+tolerance   150 mm
+
+accept      -45 0
+expect      0   -16723842.469654975459
+#roundtrip   1
+
+accept      90 0
+expect      8361921.329869444482   8361921.329869444482
+roundtrip   1
+
+accept      -90 0
+expect      -8361921.234827487729   -8361921.234827487729
+roundtrip   1
+
+accept      135 0
+expect      0.095041956369   16723842.564696932212
+roundtrip   1
+
+accept      -135 0
+expect      -16723842.430287310854   -0.039367665210
+#roundtrip   1
+
+accept      179.99 0
+expect      -8360808.039828131907   8363034.429826845415
+#roundtrip   1
+
+accept      -179.99 0
+expect      -8363034.429826845415   8360808.039828131907
+#roundtrip   1
+
+accept      45 45
+expect      5299570.257319082506   0
+roundtrip   1
+
+accept      -45 45
+expect      0   -5299570.257319079712
+roundtrip   1
+
+accept      90 45
+expect      3725360.212758610491   3725360.212758610491
+roundtrip   1
+
+accept      -90 45
+expect      -3725360.212758613285   -3725360.212758613285
+roundtrip   1
+
+accept      135 45
+expect      0   5299570.257319079712
+roundtrip   1
+
+accept      -135 45
+expect      -5299570.257319079712   0
+#roundtrip   1
+
+accept      179.99 45
+expect      -3724717.456456150394   3726002.863303491380
+roundtrip   1
+
+accept      -179.99 45
+expect      -3726002.863303492777   3724717.456456151791
+roundtrip   1
+
+accept      45 -45
+expect      16723842.303160080686   11424272.045840997249
+roundtrip   1
+
+accept      -45 -45
+expect      11424272.045840999112   -16723842.303160080686
+roundtrip   1
+
+accept      90 -45
+expect      12998482.090401468799   12998482.090401468799
+roundtrip   1
+
+accept      -90 -45
+expect      -12998482.090401465073   -12998482.090401465073
+roundtrip   1
+
+accept      135 -45
+expect      -11424272.045840999112   16723842.303160080686
+roundtrip   1
+
+accept      -135 -45
+expect      -16723842.303160080686   -11424272.045840999112
+roundtrip   1
+
+accept      179.99 -45
+expect      -12997839.439856586978   12999124.846703927964
+roundtrip   1
+
+accept      -179.99 -45
+expect      -12999124.846703927964   12997839.439856585115
+roundtrip   1
+
+accept      45 -89.999
+expect      16723842.303160080686   16723730.983657168224
+#roundtrip   1
+
+accept      -45 -89.999
+expect      16723730.983657168224   -16723842.303160080686
+#roundtrip   1
+
+accept      90 -89.999
+expect      16723763.588384689763   16723763.588384689763
+roundtrip   1
+
+accept      -90 -89.999
+expect      -16723763.588384689763   -16723763.588384689763
+roundtrip   1
+
+accept      135 -89.999
+expect      -16723730.983657168224   16723842.303160080686
+#roundtrip   1
+
+accept      -135 -89.999
+expect      -16723842.303160080686   -16723730.983657168224
+#roundtrip   1
+
+accept      179.99 -89.999
+expect      -16723763.588384689763   16723763.588384689763
+#roundtrip   1
+
+accept      -179.99 -89.999
+expect      -16723763.588384689763   16723763.588384689763
+#roundtrip   1
+
+# Test inverse
+------------------------------------------------------------
+operation  +proj=peirce_q +shape=diamond
+------------------------------------------------------------
+
+#tolerance   1 mm
+# has to bump to this for i386
+tolerance   150 mm
+
+accept      0 90
+expect      0 0
+roundtrip   1
+
+accept      0 -90
+#tolerance   10 mm
+expect      0 -23651084.600117880851
+roundtrip   1
+#tolerance   1 mm
+
+accept      0 45
+expect      0 -5268454.937608348206
+roundtrip   1
+
+accept      0 -45
+expect      0 -18382629.662509534508
+roundtrip   1
+
+accept      45 0
+tolerance   200 mm
+expect      11825542.417788611725   -11825542.552198234946
+roundtrip   1
+tolerance   150 mm
+
+accept      -45 0
+expect      -11825542.417788611725   -11825542.417788611725
+roundtrip   1
+
+accept      90 0
+expect      11825542.552198234946   0.000000000000
+roundtrip   1
+
+accept      -90 0
+expect      -11825542.417788611725   0.000000000000
+#tolerance   20 mm
+#roundtrip   1
+#tolerance   1 mm
+
+accept      135 0
+expect      11825542.552198234946   11825542.417788611725
+roundtrip   1
+
+accept      -135 0
+expect      -11825542.417788611725   11825542.362114325166
+roundtrip   1
+
+accept      179.99 0
+expect      1574.295465656175   11825542.417788611725
+#tolerance   200 mm
+#roundtrip   1
+#tolerance   1 mm
+
+accept      -179.99 0
+expect      -1574.295465656175   11825542.417788611725
+#tolerance   30 mm
+#roundtrip   1
+#tolerance   1 mm
+
+accept      45 45
+expect      3747362.066324858926   -3747362.066324859392
+roundtrip   1
+
+accept      -45 45
+expect      -3747362.066324857529   -3747362.066324857995
+roundtrip   1
+
+accept      90 45
+expect      5268454.937608345412   0.000000000000
+#roundtrip   1
+
+accept      -90 45
+expect      -5268454.937608350068   0.000000000000
+#roundtrip   1
+
+accept      135 45
+expect      3747362.066324858926   3747362.066324857995
+roundtrip   1
+
+accept      -135 45
+expect      -3747362.066324857529   3747362.066324857529
+roundtrip   1
+
+accept      179.99 45
+expect      908.919898338959   5268454.862826444209
+roundtrip   1
+
+accept      -179.99 45
+expect      -908.919898338959   5268454.862826446071
+roundtrip   1
+
+accept      45 -45
+expect      19903722.533793020993   -3747362.066324859392
+roundtrip   1
+
+accept      -45 -45
+expect      -3747362.066324857529   -19903722.533793020993
+roundtrip   1
+
+accept      90 -45
+expect      18382629.662509534508   0.000000000000
+roundtrip   1
+
+accept      -90 -45
+expect      -18382629.662509530783   0.000000000000
+#tolerance   3 mm
+#roundtrip   1
+#tolerance   1 mm
+
+accept      135 -45
+expect      3747362.066324858926   19903722.533793020993
+roundtrip   1
+
+accept      -135 -45
+expect      -19903722.533793020993   3747362.066324857529
+roundtrip   1
+
+accept      179.99 -45
+expect      908.919898338959   18382629.737291436642
+#roundtrip   1
+
+accept      -179.99 -45
+expect      -908.919898338959   18382629.737291432917
+roundtrip   1
+
+accept      45 -89.999
+expect      23651005.885342493653   -78.714775386137
+#roundtrip   1
+
+accept      -45 -89.999
+expect      -78.714775386137   -23651005.885342493653
+#roundtrip   1
+
+accept      90 -89.999
+expect      23650973.280614964664   0.000000000000
+#roundtrip   1
+
+accept      -90 -89.999
+expect      -23650973.280614964664   0.000000000000
+#roundtrip   1
+
+accept      135 -89.999
+expect      78.714775386137   23651005.885342493653
+#roundtrip   1
+
+accept      -135 -89.999
+expect      -23651005.885342493653   78.714775386137
+#roundtrip   1
+
+accept      179.99 -89.999
+expect      0.000000000000   23650973.280614964664
+#roundtrip   1
+
+accept      -179.99 -89.999
+expect      0.000000000000   23650973.280614964664
+#roundtrip   1
+
 </gie-strict>
diff --git a/test/unit/test_io.cpp b/test/unit/test_io.cpp
index 5ffa461b2d..1c7fc81296 100644
--- a/test/unit/test_io.cpp
+++ b/test/unit/test_io.cpp
@@ -6075,6 +6075,38 @@ static const struct {
          {"False northing", 2},
      }},
 
+    {"Peirce_Quincuncial",
+     {{"False_Easting", 1},
+      {"False_Northing", 2},
+      {"Central_Meridian", 3},
+      {"Scale_Factor", 4},
+      {"Latitude_Of_Origin", 5},
+      {"Option", 0}},
+     "Peirce Quincuncial (Square)",
+     {
+         {"Latitude of natural origin", 5},
+         {"Longitude of natural origin", 3},
+         {"Scale factor at natural origin", 4},
+         {"False easting", 1},
+         {"False northing", 2},
+     }},
+
+    {"Peirce_Quincuncial",
+     {{"False_Easting", 1},
+      {"False_Northing", 2},
+      {"Central_Meridian", 3},
+      {"Scale_Factor", 4},
+      {"Latitude_Of_Origin", 5},
+      {"Option", 1}},
+     "Peirce Quincuncial (Diamond)",
+     {
+         {"Latitude of natural origin", 5},
+         {"Longitude of natural origin", 3},
+         {"Scale factor at natural origin", 4},
+         {"False easting", 1},
+         {"False northing", 2},
+     }},
+
     {
         "Unknown_Method",
         {{"False_Easting", 1},
@@ -9931,6 +9963,49 @@ TEST(io, projparse_ortho_spherical_on_sphere) {
 
 // ---------------------------------------------------------------------------
 
+TEST(io, projparse_peirce_q) {
+    std::string input("+proj=peirce_q +shape=square +type=crs");
+    auto obj = PROJStringParser().createFromPROJString(input);
+    auto crs = nn_dynamic_pointer_cast<ProjectedCRS>(obj);
+    ASSERT_TRUE(crs != nullptr);
+    EXPECT_EQ(
+        crs->exportToPROJString(
+            PROJStringFormatter::create(PROJStringFormatter::Convention::PROJ_4)
+                .get()),
+        "+proj=peirce_q +shape=square +lat_0=90 +lon_0=0 +k_0=1 +x_0=0 +y_0=0 "
+        "+datum=WGS84 +units=m +no_defs +type=crs");
+}
+
+// ---------------------------------------------------------------------------
+
+TEST(io, projparse_peirce_q_invalid_lat_0) {
+    std::string input("+proj=peirce_q +lat_0=0 +shape=square +type=crs");
+    auto obj = PROJStringParser().createFromPROJString(input);
+    auto crs = nn_dynamic_pointer_cast<ProjectedCRS>(obj);
+    ASSERT_TRUE(crs != nullptr);
+    EXPECT_THROW(
+        crs->exportToPROJString(
+            PROJStringFormatter::create(PROJStringFormatter::Convention::PROJ_4)
+                .get()),
+        FormattingException);
+}
+
+// ---------------------------------------------------------------------------
+
+TEST(io, projparse_peirce_q_invalid_k_0) {
+    std::string input("+proj=peirce_q +k_0=0.5 +shape=square +type=crs");
+    auto obj = PROJStringParser().createFromPROJString(input);
+    auto crs = nn_dynamic_pointer_cast<ProjectedCRS>(obj);
+    ASSERT_TRUE(crs != nullptr);
+    EXPECT_THROW(
+        crs->exportToPROJString(
+            PROJStringFormatter::create(PROJStringFormatter::Convention::PROJ_4)
+                .get()),
+        FormattingException);
+}
+
+// ---------------------------------------------------------------------------
+
 TEST(io, projparse_axisswap_unitconvert_longlat_proj) {
     std::string input =
         "+type=crs +proj=pipeline +step +proj=axisswap +order=2,1 +step "