BUG: DatetimeIndex.is_year_start and DatetimeIndex.is_quarter_start always return False on double-digit frequencies

doc/source/whatsnew/v3.0.0.rst

@@ -410,6 +410,7 @@ Interval
 Indexing
 ^^^^^^^^
 - Bug in :meth:`DataFrame.__getitem__` returning modified columns when called with ``slice`` in Python 3.12 (:issue:`57500`)
+- Bug in :meth:`DatetimeIndex.is_year_start` and :meth:`DatetimeIndex.is_quarter_start` returning False on double-digit frequencies (:issue:`58523`)
 -
 
 Missing

pandas/_libs/tslibs/fields.pyx

@@ -3,6 +3,7 @@ Functions for accessing attributes of Timestamp/datetime64/datetime-like
 objects and arrays
 """
 from locale import LC_TIME
+import re
 
 from _strptime import LocaleTime
 
@@ -253,8 +254,7 @@ def get_start_end_field(
         # month of year. Other offsets use month, startingMonth as ending
         # month of year.
 
-        if (freqstr[0:2] in ["MS", "QS", "YS"]) or (
-                freqstr[1:3] in ["MS", "QS", "YS"]):
+        if re.split("[0-9]*", freqstr, maxsplit=1)[1][0:2] in ["MS", "QS", "YS"]:
             end_month = 12 if month_kw == 1 else month_kw - 1
             start_month = month_kw
         else:

pandas/tests/indexes/datetimes/test_scalar_compat.py

@@ -328,3 +328,11 @@ def test_dti_is_month_start_custom(self):
         msg = "Custom business days is not supported by is_month_start"
         with pytest.raises(ValueError, match=msg):
             dti.is_month_start
+
+    def test_dti_is_year_quarter_start_doubledigit_freq(self):
+        # GH#58523
+        dr = date_range("2017-01-01", periods=2, freq="10YS")
+        assert all(dr.is_year_start)
+
+        dr = date_range("2017-01-01", periods=2, freq="10QS")
+        assert all(dr.is_quarter_start)