diff --git a/LICENSE/LICENSE_GNU_FREEFONT b/LICENSE/LICENSE_GNU_FREEFONT
diff --git a/extern/ttconv/pprdrv_tt.cpp b/extern/ttconv/pprdrv_tt.cpp
@@ -107,6 +107,61 @@ Fixed getFixed(BYTE *s)
     return val;
 } /* end of getFixed() */
 
+/*
+** Get a 16 bit fixed point (2.14) number.
+**
+** The value range is from -2.0 to a little under 2.0.  The two
+** most-significant bits encode the whole part in two's-complement
+** format and the remaining 14 bits encode the numerator n of the
+** fraction n/16384 to be added to the whole part.
+**
+** https://docs.microsoft.com/en-us/typography/opentype/otspec181/otff#data-types
+*/
+F2DOT14 getF2DOT14(BYTE *s)
+{
+    F2DOT14 val={0, 0};
+
+    int sign = s[0] & 0x80;
+    int bit = (s[0] & 0x40) >> 6;
+    if (sign) {
+      val.whole = bit-2;
+    } else {
+      val.whole = bit;
+    }
+    val.fraction = ((s[0] & 0x3f) << 8) + s[1];
+
+    return val;
+}
+
+/*
+** Convert a fixed-point 2.14 number to a newly-allocated string.
+** Use at most six decimals but remove trailing zeroes and possibly
+** the decimal point.
+*/
+char* F2DOT14value(F2DOT14 f)
+{
+    const size_t maxlen = sizeof("-1.234567");
+    char *value = (char*)calloc(sizeof(char), maxlen);
+    if (value == NULL ||
+        snprintf(value, maxlen, "%.6f", f.whole + (float)f.fraction/16384) < 0) {
+        abort();
+    }
+
+    char *ptr = &value[maxlen-1];
+    while (ptr > value && *ptr == '\0') {
+        ptr--;
+    }
+    while (ptr > value && *ptr == '0') {
+        *ptr-- = '\0';
+    }
+    if (ptr > value && *ptr == '.') {
+        *ptr = '\0';
+    }
+
+    return value;
+}
+
+
 /*-----------------------------------------------------------------------
 ** Load a TrueType font table into memory and return a pointer to it.
 ** The font's "file" and "offset_table" fields must be set before this

diff --git a/extern/ttconv/pprdrv_tt2.cpp b/extern/ttconv/pprdrv_tt2.cpp
@@ -409,14 +409,17 @@ void GlyphToType3::load_char(TTFONT* font, BYTE *glyph)
 } /* end of load_char() */
 
 /*
-** Emmit PostScript code for a composite character.
+** Emit PostScript code for a composite character.
+**
+** https://docs.microsoft.com/en-us/typography/opentype/spec/glyf#composite-glyph-description
 */
 void GlyphToType3::do_composite(TTStreamWriter& stream, struct TTFONT *font, BYTE *glyph)
 {
     USHORT flags;
     USHORT glyphIndex;
     int arg1;
     int arg2;
+    F2DOT14 mat00 = {1, 0}, mat01 = {0, 0}, mat10 = {0, 0}, mat11 = {1, 0};
 
     /* Once around this loop for each component. */
     do
@@ -444,14 +447,21 @@ void GlyphToType3::do_composite(TTStreamWriter& stream, struct TTFONT *font, BYT
 
         if (flags & WE_HAVE_A_SCALE)
         {
+            mat00 = mat11 = getF2DOT14(glyph);
             glyph += 2;
         }
         else if (flags & WE_HAVE_AN_X_AND_Y_SCALE)
         {
+            mat00 = getF2DOT14(glyph);
+            mat11 = getF2DOT14(glyph+2);
             glyph += 4;
         }
         else if (flags & WE_HAVE_A_TWO_BY_TWO)
         {
+            mat00 = getF2DOT14(glyph);
+            mat01 = getF2DOT14(glyph+2);
+            mat10 = getF2DOT14(glyph+4);
+            mat11 = getF2DOT14(glyph+6);
             glyph += 8;
         }
         else
@@ -472,7 +482,15 @@ void GlyphToType3::do_composite(TTStreamWriter& stream, struct TTFONT *font, BYT
                    subglyph here.  However, that doesn't seem to work with
                    xpdf or gs (only acrobat), so instead, this just includes
                    the subglyph here inline. */
-                stream.printf("q 1 0 0 1 %d %d cm\n", topost(arg1), topost(arg2));
+                char *s00 = F2DOT14value(mat00), *s01 = F2DOT14value(mat01),
+                    *s10 = F2DOT14value(mat10), *s11 = F2DOT14value(mat11);
+
+                stream.printf("q %s %s %s %s %d %d cm\n",
+                              s00, s01, s10, s11, topost(arg1), topost(arg2));
+                free(s00);
+                free(s01);
+                free(s10);
+                free(s11);
             }
             else
             {

diff --git a/extern/ttconv/truetype.h b/extern/ttconv/truetype.h
@@ -41,6 +41,9 @@ typedef struct
     unsigned short int fraction;
     } Fixed;
 
+/* A 2.14 fixed point number */
+typedef Fixed F2DOT14;
+
 /* This structure tells what we have found out about */
 /* the current font. */
 struct TTFONT
@@ -86,6 +89,8 @@ struct TTFONT
 ULONG getULONG(BYTE *p);
 USHORT getUSHORT(BYTE *p);
 Fixed getFixed(BYTE *p);
+F2DOT14 getF2DOT14(BYTE *p);
+char *F2DOT14value(F2DOT14 f);
 
 /*
 ** Get an funits word.

diff --git a/lib/matplotlib/tests/FreeSerifSubset.ttf b/lib/matplotlib/tests/FreeSerifSubset.ttf
diff --git a/lib/matplotlib/tests/baseline_images/test_ttconv/ttconv_transforms.pdf b/lib/matplotlib/tests/baseline_images/test_ttconv/ttconv_transforms.pdf
diff --git a/lib/matplotlib/tests/test_ttconv.py b/lib/matplotlib/tests/test_ttconv.py
@@ -15,3 +15,18 @@ def test_truetype_conversion(recwarn):
             font=Path(__file__).with_name("mpltest.ttf"), fontsize=80)
     ax.set_xticks([])
     ax.set_yticks([])
+
+
+@image_comparison(["ttconv_transforms.pdf"])
+def test_ttconv_transforms():
+    matplotlib.rcParams['pdf.fonttype'] = 3
+    fig, ax = plt.subplots()
+    kw = {'font': Path(__file__).with_name("FreeSerifSubset.ttf"),
+          'fontsize': 14}
+    # characters where Free Serif uses various scales and linear transforms
+    # e.g. the right paren is a reflected left paren
+    ax.text(.1, .1, "parens, FAX, u with two diacritics: ()℻ǘ", **kw)
+    ax.text(.1, .2, "double arrows LURD, single arrows LURD: ⇐⇑⇒⇓←↑→↓", **kw)
+    ax.text(.1, .3, "corner arrows, wreath product: ↴↵≀", **kw)
+    ax.set_xticks([])
+    ax.set_yticks([])
diff --git a/setupext.py b/setupext.py
@@ -455,6 +455,7 @@ def get_package_data(self):
                 *_pkg_data_helper('matplotlib', 'tests/tinypages'),
                 'tests/cmr10.pfb',
                 'tests/mpltest.ttf',
+                'tests/FreeSerifSubset.ttf',
             ],
             'mpl_toolkits': [
                 *_pkg_data_helper('mpl_toolkits', 'tests/baseline_images'),