diff --git a/Lib/fontTools/ttLib/tables/_g_l_y_f.py b/Lib/fontTools/ttLib/tables/_g_l_y_f.py
index 095bd3177..eaa9920f0 100644
--- a/Lib/fontTools/ttLib/tables/_g_l_y_f.py
+++ b/Lib/fontTools/ttLib/tables/_g_l_y_f.py
@@ -1206,9 +1206,7 @@ class Glyph(object):
         Return True if bounds were calculated, False otherwise.
         """
         for compo in self.components:
-            if hasattr(compo, "firstPt") or hasattr(compo, "transform"):
-                return False
-            if not float(compo.x).is_integer() or not float(compo.y).is_integer():
+            if not compo._hasOnlyIntegerTranslate():
                 return False
 
         # All components are untransformed and have an integer x/y translate
@@ -1241,7 +1239,7 @@ class Glyph(object):
         else:
             return self.numberOfContours == -1
 
-    def getCoordinates(self, glyfTable, round=noRound):
+    def getCoordinates(self, glyfTable, *, round=noRound):
         """Return the coordinates, end points and flags
 
         This method returns three values: A :py:class:`GlyphCoordinates` object,
@@ -1258,11 +1256,7 @@ class Glyph(object):
         """
 
         if self.numberOfContours > 0:
-            coordinates = self.coordinates
-            if round is not noRound:
-                coordinates = GlyphCoordinates(coordinates)
-                coordinates.toInt(round=round)
-            return coordinates, self.endPtsOfContours, self.flags
+            return self.coordinates, self.endPtsOfContours, self.flags
         elif self.isComposite():
             # it's a composite
             allCoords = GlyphCoordinates()
@@ -1280,6 +1274,18 @@ class Glyph(object):
                         % compo.glyphName
                     )
                 coordinates = GlyphCoordinates(coordinates)
+                # if asked to round e.g. while computing bboxes, it's important we
+                # do it immediately before a component transform is applied to a
+                # simple glyph's coordinates in case these might still contain floats;
+                # however, if the referenced component glyph is another composite, we
+                # must not round here but only at the end, after all the nested
+                # transforms have been applied, or else rounding errors will compound.
+                if (
+                    round is not noRound
+                    and g.numberOfContours > 0
+                    and not compo._hasOnlyIntegerTranslate()
+                ):
+                    coordinates.toInt(round=round)
                 if hasattr(compo, "firstPt"):
                     # component uses two reference points: we apply the transform _before_
                     # computing the offset between the points
@@ -1936,6 +1942,18 @@ class GlyphComponent(object):
         result = self.__eq__(other)
         return result if result is NotImplemented else not result
 
+    def _hasOnlyIntegerTranslate(self):
+        """Return True if it's a 'simple' component.
+
+        That is, it has no anchor points and no transform other than integer translate.
+        """
+        return (
+            not hasattr(self, "firstPt")
+            and not hasattr(self, "transform")
+            and float(self.x).is_integer()
+            and float(self.y).is_integer()
+        )
+
 
 class GlyphCoordinates(object):
     """A list of glyph coordinates.