diff --git a/doc/source/whatsnew/v3.0.0.rst b/doc/source/whatsnew/v3.0.0.rst
index ce44050a99a79..ade659d6e9e34 100644
--- a/doc/source/whatsnew/v3.0.0.rst
+++ b/doc/source/whatsnew/v3.0.0.rst
@@ -421,6 +421,7 @@ 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
diff --git a/pandas/_libs/tslibs/fields.pyi b/pandas/_libs/tslibs/fields.pyi
index c6cfd44e9f6ab..bc55e34f3d208 100644
--- a/pandas/_libs/tslibs/fields.pyi
+++ b/pandas/_libs/tslibs/fields.pyi
@@ -16,7 +16,7 @@ def get_date_name_field(
 def get_start_end_field(
     dtindex: npt.NDArray[np.int64],
     field: str,
-    freqstr: str | None = ...,
+    freq_name: str | None = ...,
     month_kw: int = ...,
     reso: int = ...,  # NPY_DATETIMEUNIT
 ) -> npt.NDArray[np.bool_]: ...
diff --git a/pandas/_libs/tslibs/fields.pyx b/pandas/_libs/tslibs/fields.pyx
index ff4fb4d635d17..8f8060b2a5f83 100644
--- a/pandas/_libs/tslibs/fields.pyx
+++ b/pandas/_libs/tslibs/fields.pyx
@@ -210,7 +210,7 @@ cdef bint _is_on_month(int month, int compare_month, int modby) noexcept nogil:
 def get_start_end_field(
     const int64_t[:] dtindex,
     str field,
-    str freqstr=None,
+    str freq_name=None,
     int month_kw=12,
     NPY_DATETIMEUNIT reso=NPY_FR_ns,
 ):
@@ -223,7 +223,7 @@ def get_start_end_field(
     ----------
     dtindex : ndarray[int64]
     field : str
-    frestr : str or None, default None
+    freq_name : str or None, default None
     month_kw : int, default 12
     reso : NPY_DATETIMEUNIT, default NPY_FR_ns
 
@@ -243,18 +243,17 @@ def get_start_end_field(
 
     out = np.zeros(count, dtype="int8")
 
-    if freqstr:
-        if freqstr == "C":
+    if freq_name:
+        if freq_name == "C":
             raise ValueError(f"Custom business days is not supported by {field}")
-        is_business = freqstr[0] == "B"
+        is_business = freq_name[0] == "B"
 
         # YearBegin(), BYearBegin() use month = starting month of year.
         # QuarterBegin(), BQuarterBegin() use startingMonth = starting
         # 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 freq_name.lstrip("B")[0:2]  in ["MS", "QS", "YS"]:
             end_month = 12 if month_kw == 1 else month_kw - 1
             start_month = month_kw
         else:
diff --git a/pandas/_libs/tslibs/timestamps.pyx b/pandas/_libs/tslibs/timestamps.pyx
index 7e499d219243d..0010497425c02 100644
--- a/pandas/_libs/tslibs/timestamps.pyx
+++ b/pandas/_libs/tslibs/timestamps.pyx
@@ -579,15 +579,15 @@ cdef class _Timestamp(ABCTimestamp):
         if freq:
             kwds = freq.kwds
             month_kw = kwds.get("startingMonth", kwds.get("month", 12))
-            freqstr = freq.freqstr
+            freq_name = freq.name
         else:
             month_kw = 12
-            freqstr = None
+            freq_name = None
 
         val = self._maybe_convert_value_to_local()
 
         out = get_start_end_field(np.array([val], dtype=np.int64),
-                                  field, freqstr, month_kw, self._creso)
+                                  field, freq_name, month_kw, self._creso)
         return out[0]
 
     @property
diff --git a/pandas/core/arrays/datetimes.py b/pandas/core/arrays/datetimes.py
index 3961b0a7672f4..b075e3d299ed0 100644
--- a/pandas/core/arrays/datetimes.py
+++ b/pandas/core/arrays/datetimes.py
@@ -145,8 +145,12 @@ def f(self):
                     kwds = freq.kwds
                     month_kw = kwds.get("startingMonth", kwds.get("month", 12))
 
+                if freq is not None:
+                    freq_name = freq.name
+                else:
+                    freq_name = None
                 result = fields.get_start_end_field(
-                    values, field, self.freqstr, month_kw, reso=self._creso
+                    values, field, freq_name, month_kw, reso=self._creso
                 )
             else:
                 result = fields.get_date_field(values, field, reso=self._creso)
diff --git a/pandas/tests/indexes/datetimes/test_scalar_compat.py b/pandas/tests/indexes/datetimes/test_scalar_compat.py
index f766894a993a0..1f2cfe8b7ddc8 100644
--- a/pandas/tests/indexes/datetimes/test_scalar_compat.py
+++ b/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)