BUG: Make DTI/TDI/PI argsort match their underlying arrays

pandas/core/groupby/grouper.py

@@ -374,7 +374,10 @@ def _set_grouper(self, obj: FrameOrSeries, sort: bool = False):
         # possibly sort
         if (self.sort or sort) and not ax.is_monotonic:
             # use stable sort to support first, last, nth
-            indexer = self.indexer = ax.argsort(kind="mergesort")
+            # TODO: why does putting na_position="first" fix datetimelike cases?
+            indexer = self.indexer = ax.array.argsort(
+                kind="mergesort", na_position="first"
+            )
             ax = ax.take(indexer)
             obj = obj.take(indexer, axis=self.axis)
 

pandas/core/indexes/base.py

@@ -4776,10 +4776,6 @@ def argsort(self, *args, **kwargs) -> np.ndarray:
         >>> idx[order]
         Index(['a', 'b', 'c', 'd'], dtype='object')
         """
-        if needs_i8_conversion(self.dtype):
-            # TODO: these do not match the underlying EA argsort methods GH#37863
-            return self.asi8.argsort(*args, **kwargs)
-
         # This works for either ndarray or EA, is overriden
         #  by RangeIndex, MultIIndex
         return self._data.argsort(*args, **kwargs)

pandas/tests/indexes/datetimelike.py

@@ -10,6 +10,14 @@
 
 
 class DatetimeLike(Base):
+    def test_argsort_matches_array(self):
+        rng = self.create_index()
+        rng = rng.insert(1, pd.NaT)
+
+        result = rng.argsort()
+        expected = rng._data.argsort()
+        tm.assert_numpy_array_equal(result, expected)
+
     def test_can_hold_identifiers(self):
         idx = self.create_index()
         key = idx[0]