diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index 9a723a88941b6..f8d4dd4c78bcb 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -1486,7 +1486,8 @@ def __rsub__(self, other):
             # GH#19959 datetime - datetime is well-defined as timedelta,
             # but any other type - datetime is not well-defined.
             raise TypeError(
-                f"cannot subtract {type(self).__name__} from {type(other).__name__}"
+                f"cannot subtract {type(self).__name__} from "
+                f"{type(other).__name__}[{other.dtype}]"
             )
         elif isinstance(self.dtype, PeriodDtype) and lib.is_np_dtype(other_dtype, "m"):
             # TODO: Can we simplify/generalize these cases at all?
@@ -1495,8 +1496,14 @@ def __rsub__(self, other):
             self = cast("TimedeltaArray", self)
             return (-self) + other
 
+        flipped = self - other
+        if flipped.dtype.kind == "M":
+            # GH#59571 give a more helpful exception message
+            raise TypeError(
+                f"cannot subtract {type(self).__name__} from {type(other).__name__}"
+            )
         # We get here with e.g. datetime objects
-        return -(self - other)
+        return -flipped
 
     def __iadd__(self, other) -> Self:
         result = self + other
diff --git a/pandas/tests/arithmetic/test_datetime64.py b/pandas/tests/arithmetic/test_datetime64.py
index d439ff723b355..5d831e0ec25a1 100644
--- a/pandas/tests/arithmetic/test_datetime64.py
+++ b/pandas/tests/arithmetic/test_datetime64.py
@@ -955,7 +955,12 @@ def test_dt64arr_add_sub_td64ndarray(self, tz_naive_fixture, box_with_array):
 
         result = dtarr - tdarr
         tm.assert_equal(result, expected)
-        msg = "cannot subtract|(bad|unsupported) operand type for unary"
+        msg = "|".join(
+            [
+                "cannot subtract DatetimeArray from ndarray",
+                "cannot subtract a datelike from a TimedeltaArray",
+            ]
+        )
         with pytest.raises(TypeError, match=msg):
             tdarr - dtarr
 
@@ -1272,7 +1277,7 @@ def test_dt64arr_series_sub_tick_DateOffset(self, box_with_array):
 
         result2 = -pd.offsets.Second(5) + ser
         tm.assert_equal(result2, expected)
-        msg = "(bad|unsupported) operand type for unary"
+        msg = "cannot subtract DatetimeArray from Second"
         with pytest.raises(TypeError, match=msg):
             pd.offsets.Second(5) - ser
 
@@ -1317,9 +1322,7 @@ def test_dti_add_tick_tzaware(self, tz_aware_fixture, box_with_array):
             roundtrip = offset - scalar
             tm.assert_equal(roundtrip, dates)
 
-            msg = "|".join(
-                ["bad operand type for unary -", "cannot subtract DatetimeArray"]
-            )
+            msg = "cannot subtract DatetimeArray from"
             with pytest.raises(TypeError, match=msg):
                 scalar - dates
 
@@ -1378,7 +1381,7 @@ def test_dt64arr_add_sub_relativedelta_offsets(self, box_with_array, unit):
             expected = DatetimeIndex([x - off for x in vec_items]).as_unit(exp_unit)
             expected = tm.box_expected(expected, box_with_array)
             tm.assert_equal(expected, vec - off)
-            msg = "(bad|unsupported) operand type for unary"
+            msg = "cannot subtract DatetimeArray from"
             with pytest.raises(TypeError, match=msg):
                 off - vec
 
@@ -1494,7 +1497,7 @@ def test_dt64arr_add_sub_DateOffsets(
         expected = DatetimeIndex([offset + x for x in vec_items]).as_unit(unit)
         expected = tm.box_expected(expected, box_with_array)
         tm.assert_equal(expected, offset + vec)
-        msg = "(bad|unsupported) operand type for unary"
+        msg = "cannot subtract DatetimeArray from"
         with pytest.raises(TypeError, match=msg):
             offset - vec
 
@@ -1983,7 +1986,7 @@ def test_operators_datetimelike_with_timezones(self):
         result = dt1 - td1[0]
         exp = (dt1.dt.tz_localize(None) - td1[0]).dt.tz_localize(tz)
         tm.assert_series_equal(result, exp)
-        msg = "(bad|unsupported) operand type for unary"
+        msg = "cannot subtract DatetimeArray from"
         with pytest.raises(TypeError, match=msg):
             td1[0] - dt1
 
diff --git a/pandas/tests/arithmetic/test_timedelta64.py b/pandas/tests/arithmetic/test_timedelta64.py
index 87e085fb22878..642420713aeba 100644
--- a/pandas/tests/arithmetic/test_timedelta64.py
+++ b/pandas/tests/arithmetic/test_timedelta64.py
@@ -320,7 +320,7 @@ def test_subtraction_ops(self):
         with pytest.raises(TypeError, match=msg):
             td - dt
 
-        msg = "(bad|unsupported) operand type for unary"
+        msg = "cannot subtract DatetimeArray from Timedelta"
         with pytest.raises(TypeError, match=msg):
             td - dti