diff --git a/doc/api/next_api_changes/deprecations/29817-AL.rst b/doc/api/next_api_changes/deprecations/29817-AL.rst
index 204751d1ea85..f3b339ed7c10 100644
--- a/doc/api/next_api_changes/deprecations/29817-AL.rst
+++ b/doc/api/next_api_changes/deprecations/29817-AL.rst
@@ -1,3 +1,7 @@
 ``DviFont.widths``
 ~~~~~~~~~~~~~~~~~~
 ... is deprecated with no replacement.
+
+Direct access to ``Tfm``'s ``widths``, ``heights``, ``depths`` dicts
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+... is deprecated; access a glyph's metrics with `.Tfm.get_metrics` instead.
diff --git a/lib/matplotlib/backends/backend_pdf.py b/lib/matplotlib/backends/backend_pdf.py
index 559a684050b8..8bb1736f3d27 100644
--- a/lib/matplotlib/backends/backend_pdf.py
+++ b/lib/matplotlib/backends/backend_pdf.py
@@ -993,8 +993,9 @@ def _embedTeXFont(self, fontinfo):
         widthsObject = self.reserveObject('font widths')
         tfm = fontinfo.dvifont._tfm
         # convert from TeX's 12.20 representation to 1/1000 text space units.
-        widths = [(1000 * tfm.width.get(char, 0)) >> 20
-                  for char in range(max(tfm.width, default=-1) + 1)]
+        widths = [(1000 * metrics.tex_width) >> 20
+                  if (metrics := tfm.get_metrics(char)) else 0
+                  for char in range(max(tfm._glyph_metrics, default=-1) + 1)]
         self.writeObject(widthsObject, widths)
 
         # Font dictionary
diff --git a/lib/matplotlib/dviread.py b/lib/matplotlib/dviread.py
index 1e6dcfdc4409..3f05e1cf0c80 100644
--- a/lib/matplotlib/dviread.py
+++ b/lib/matplotlib/dviread.py
@@ -18,6 +18,7 @@
 """
 
 from collections import namedtuple
+import dataclasses
 import enum
 from functools import cache, lru_cache, partial, wraps
 import logging
@@ -604,32 +605,30 @@ def __repr__(self):
 
     def _width_of(self, char):
         """Width of char in dvi units."""
-        width = self._tfm.width.get(char, None)
-        if width is not None:
-            return _mul1220(width, self._scale)
-        _log.debug('No width for char %d in font %s.', char, self.texname)
-        return 0
+        metrics = self._tfm.get_metrics(char)
+        if metrics is None:
+            _log.debug('No width for char %d in font %s.', char, self.texname)
+            return 0
+        return _mul1220(metrics.tex_width, self._scale)
 
     def _height_depth_of(self, char):
         """Height and depth of char in dvi units."""
-        result = []
-        for metric, name in ((self._tfm.height, "height"),
-                             (self._tfm.depth, "depth")):
-            value = metric.get(char, None)
-            if value is None:
-                _log.debug('No %s for char %d in font %s',
-                           name, char, self.texname)
-                result.append(0)
-            else:
-                result.append(_mul1220(value, self._scale))
+        metrics = self._tfm.get_metrics(char)
+        if metrics is None:
+            _log.debug('No metrics for char %d in font %s', char, self.texname)
+            return [0, 0]
+        hd = [
+            _mul1220(metrics.tex_height, self._scale),
+            _mul1220(metrics.tex_depth, self._scale),
+        ]
         # cmsyXX (symbols font) glyph 0 ("minus") has a nonzero descent
         # so that TeX aligns equations properly
         # (https://tex.stackexchange.com/q/526103/)
         # but we actually care about the rasterization depth to align
         # the dvipng-generated images.
         if re.match(br'^cmsy\d+$', self.texname) and char == 0:
-            result[-1] = 0
-        return result
+            hd[-1] = 0
+        return hd
 
 
 class Vf(Dvi):
@@ -761,6 +760,22 @@ def _mul1220(num1, num2):
     return (num1*num2) >> 20
 
 
+@dataclasses.dataclass(frozen=True, kw_only=True)
+class TexMetrics:
+    """
+    Metrics of a glyph, with TeX semantics.
+
+    TeX metrics have different semantics from FreeType metrics: tex_width
+    corresponds to FreeType's ``advance`` (i.e., including whitespace padding);
+    tex_height to ``bearingY`` (how much the glyph extends over the baseline);
+    tex_depth to ``height - bearingY`` (how much the glyph extends under the
+    baseline, as a positive number).
+    """
+    tex_width: int
+    tex_height: int
+    tex_depth: int
+
+
 class Tfm:
     """
     A TeX Font Metric file.
@@ -778,12 +793,7 @@ class Tfm:
     design_size : int
        Design size of the font (in 12.20 TeX points); unused because it is
        overridden by the scale factor specified in the dvi file.
-    width, height, depth : dict
-       Dimensions of each character, need to be scaled by the factor
-       specified in the dvi file. These are dicts because indexing may
-       not start from 0.
     """
-    __slots__ = ('checksum', 'design_size', 'width', 'height', 'depth')
 
     def __init__(self, filename):
         _log.debug('opening tfm file %s', filename)
@@ -799,15 +809,26 @@ def __init__(self, filename):
             widths = struct.unpack(f'!{nw}i', file.read(4*nw))
             heights = struct.unpack(f'!{nh}i', file.read(4*nh))
             depths = struct.unpack(f'!{nd}i', file.read(4*nd))
-        self.width = {}
-        self.height = {}
-        self.depth = {}
+        self._glyph_metrics = {}
         for idx, char in enumerate(range(bc, ec+1)):
             byte0 = char_info[4*idx]
             byte1 = char_info[4*idx+1]
-            self.width[char] = widths[byte0]
-            self.height[char] = heights[byte1 >> 4]
-            self.depth[char] = depths[byte1 & 0xf]
+            self._glyph_metrics[char] = TexMetrics(
+                tex_width=widths[byte0],
+                tex_height=heights[byte1 >> 4],
+                tex_depth=depths[byte1 & 0xf],
+            )
+
+    def get_metrics(self, idx):
+        """Return a glyph's TexMetrics, or None if unavailable."""
+        return self._glyph_metrics.get(idx)
+
+    width = _api.deprecated("3.11", alternative="get_metrics")(
+        property(lambda self: {c: m.tex_width for c, m in self._glyph_metrics}))
+    height = _api.deprecated("3.11", alternative="get_metrics")(
+        property(lambda self: {c: m.tex_height for c, m in self._glyph_metrics}))
+    depth = _api.deprecated("3.11", alternative="get_metrics")(
+        property(lambda self: {c: m.tex_depth for c, m in self._glyph_metrics}))
 
 
 PsFont = namedtuple('PsFont', 'texname psname effects encoding filename')
diff --git a/lib/matplotlib/dviread.pyi b/lib/matplotlib/dviread.pyi
index 29d9288c047f..f8d8f979fd8c 100644
--- a/lib/matplotlib/dviread.pyi
+++ b/lib/matplotlib/dviread.pyi
@@ -1,3 +1,4 @@
+import dataclasses
 from pathlib import Path
 import io
 import os
@@ -68,13 +69,25 @@ class Vf(Dvi):
     def __init__(self, filename: str | os.PathLike) -> None: ...
     def __getitem__(self, code: int) -> Page: ...
 
+@dataclasses.dataclass(frozen=True, kw_only=True)
+class TexMetrics:
+    tex_width: int
+    tex_height: int
+    tex_depth: int
+    # work around mypy not respecting kw_only=True in stub files
+    __match_args__ = ()
+
 class Tfm:
     checksum: int
     design_size: int
-    width: dict[int, int]
-    height: dict[int, int]
-    depth: dict[int, int]
     def __init__(self, filename: str | os.PathLike) -> None: ...
+    def get_metrics(self, idx: int) -> TexMetrics | None: ...
+    @property
+    def width(self) -> dict[int, int]: ...
+    @property
+    def height(self) -> dict[int, int]: ...
+    @property
+    def depth(self) -> dict[int, int]: ...
 
 class PsFont(NamedTuple):
     texname: bytes