diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 53b5bf8f182bd..89b3fa1c6a6ce 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -3070,6 +3070,30 @@ def intersection(self, other, sort=False):
                 return self.unique()._get_reconciled_name_object(other)
             return self._get_reconciled_name_object(other)
 
+        if len(self) == 0 or len(other) == 0:
+            # fastpath; we need to be careful about having commutativity
+
+            if self._is_multi or other._is_multi:
+                # _convert_can_do_setop ensures that we have both or neither
+                # We retain self.levels
+                return self[:0].rename(result_name)
+
+            dtype = self._find_common_type_compat(other)
+            if is_dtype_equal(self.dtype, dtype):
+                # Slicing allows us to retain DTI/TDI.freq, RangeIndex
+
+                # Note: self[:0] vs other[:0] affects
+                #  1) which index's `freq` we get in DTI/TDI cases
+                #     This may be a historical artifact, i.e. no documented
+                #     reason for this choice.
+                #  2) The `step` we get in RangeIndex cases
+                if len(self) == 0:
+                    return self[:0].rename(result_name)
+                else:
+                    return other[:0].rename(result_name)
+
+            return Index([], dtype=dtype, name=result_name)
+
         elif not self._should_compare(other):
             # We can infer that the intersection is empty.
             if isinstance(self, ABCMultiIndex):
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 8da99961a0a24..0ab41465c0636 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -532,15 +532,11 @@ def is_type_compatible(self, kind: str) -> bool:
 
     def _intersection(self, other: Index, sort=False) -> Index:
         """
-        intersection specialized to the case with matching dtypes.
+        intersection specialized to the case with matching dtypes and both non-empty.
         """
         other = cast("DatetimeTimedeltaMixin", other)
-        if len(self) == 0:
-            return self.copy()._get_reconciled_name_object(other)
-        if len(other) == 0:
-            return other.copy()._get_reconciled_name_object(self)
 
-        elif not self._can_fast_intersect(other):
+        if not self._can_fast_intersect(other):
             result = Index._intersection(self, other, sort=sort)
             # We need to invalidate the freq because Index._intersection
             #  uses _shallow_copy on a view of self._data, which will preserve
@@ -550,6 +546,11 @@ def _intersection(self, other: Index, sort=False) -> Index:
             result = self._wrap_setop_result(other, result)
             return result._with_freq(None)._with_freq("infer")
 
+        else:
+            return self._fast_intersect(other, sort)
+
+    def _fast_intersect(self, other, sort):
+
         # to make our life easier, "sort" the two ranges
         if self[0] <= other[0]:
             left, right = self, other
@@ -627,11 +628,7 @@ def _can_fast_union(self: _T, other: _T) -> bool:
         return (right_start == left_end + freq) or right_start in left
 
     def _fast_union(self: _T, other: _T, sort=None) -> _T:
-        if len(other) == 0:
-            return self.view(type(self))
-
-        if len(self) == 0:
-            return other.view(type(self))
+        # Caller is responsible for ensuring self and other are non-empty
 
         # to make our life easier, "sort" the two ranges
         if self[0] <= other[0]:
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 2a2993dedb25d..3971039a09b71 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -405,7 +405,8 @@ def union_many(self, others):
 
             this, other = this._maybe_utc_convert(other)
 
-            if this._can_fast_union(other):
+            if len(self) and len(other) and this._can_fast_union(other):
+                # union already has fastpath handling for empty cases
                 this = this._fast_union(other)
             else:
                 this = Index.union(this, other)
diff --git a/pandas/core/indexes/range.py b/pandas/core/indexes/range.py
index 41997ad7ca46e..1c6ad0b3ddd30 100644
--- a/pandas/core/indexes/range.py
+++ b/pandas/core/indexes/range.py
@@ -553,14 +553,12 @@ def equals(self, other: object) -> bool:
     # Set Operations
 
     def _intersection(self, other: Index, sort=False):
+        # caller is responsible for checking self and other are both non-empty
 
         if not isinstance(other, RangeIndex):
             # Int64Index
             return super()._intersection(other, sort=sort)
 
-        if not len(self) or not len(other):
-            return self._simple_new(_empty_range)
-
         first = self._range[::-1] if self.step < 0 else self._range
         second = other._range[::-1] if other.step < 0 else other._range