diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 7e6461f0fab5e..183267efd8170 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -627,7 +627,7 @@ def _fast_intersect(self, other, sort):
 
 def _can_fast_intersect(self, other: Self) -> bool:
 # Note: we only get here with len(self)> 0 and len(other)> 0
- if self.freq is None:
+ if self.freq is None or self.freq == "C":
 return False
 
 elif other.freq != self.freq:
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index 7a68cb867c94e..20920c131232a 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -7,6 +7,7 @@
 import numpy as np
 import pytest
 
+from pandas._libs.tslibs.timedeltas import Timedelta
 import pandas.util._test_decorators as td
 
 import pandas as pd
@@ -709,7 +710,6 @@ def test_intersection_bug(self):
 b = bdate_range("12/10/2011", "12/20/2011", freq="C")
 result = a.intersection(b)
 tm.assert_index_equal(result, b)
- assert result.freq == b.freq
 
 @pytest.mark.parametrize(
 "tz", [None, "UTC", "Europe/Berlin", timezone(timedelta(hours=-1))]
@@ -757,3 +757,16 @@ def test_intersection_non_nano_rangelike():
 freq="D",
 )
 tm.assert_index_equal(result, expected)
+
+
+def test_cday_intersection_empty():
+ # GH#44025
+ off = pd.offsets.CDay(1, normalize=False)
+ ts = Timestamp("2021-10-13 09:00")
+ ts2 = ts + Timedelta("1 hour")
+
+ dti1 = date_range(start=ts, periods=10, freq=off)
+ dti2 = date_range(start=ts2, periods=10, freq=off)
+ result = dti1.intersection(dti2)
+ expected = DatetimeIndex([], dtype="datetime64[ns]")
+ tm.assert_index_equal(result, expected)
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