diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 88bf0e005a221..16426e11c5a24 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -759,6 +759,7 @@ Datetimelike
 - Bug in :meth:`DatetimeIndex.to_period` not infering the frequency when called with no arguments (:issue:`33358`)
 - Bug in :meth:`DatetimeIndex.tz_localize` incorrectly retaining ``freq`` in some cases where the original freq is no longer valid (:issue:`30511`)
 - Bug in :meth:`DatetimeIndex.intersection` losing ``freq`` and timezone in some cases (:issue:`33604`)
+- Bug in :meth:`DatetimeIndex.get_indexer` where incorrect output would be returned for mixed datetime-like targets (:issue:`33741`)
 - Bug in :class:`DatetimeIndex` addition and subtraction with some types of :class:`DateOffset` objects incorrectly retaining an invalid ``freq`` attribute (:issue:`33779`)
 - Bug in :class:`DatetimeIndex` where setting the ``freq`` attribute on an index could silently change the ``freq`` attribute on another index viewing the same data (:issue:`33552`)
 - :meth:`DataFrame.min`/:meth:`DataFrame.max` not returning consistent result with :meth:`Series.min`/:meth:`Series.max` when called on objects initialized with empty :func:`pd.to_datetime`
diff --git a/pandas/_libs/lib.pyx b/pandas/_libs/lib.pyx
index 222b7af4e4b1c..ea97bab2198eb 100644
--- a/pandas/_libs/lib.pyx
+++ b/pandas/_libs/lib.pyx
@@ -1380,8 +1380,10 @@ def infer_dtype(value: object, skipna: bool = True) -> str:
         return "mixed-integer"
 
     elif PyDateTime_Check(val):
-        if is_datetime_array(values):
+        if is_datetime_array(values, skipna=skipna):
             return "datetime"
+        elif is_date_array(values, skipna=skipna):
+            return "date"
 
     elif PyDate_Check(val):
         if is_date_array(values, skipna=skipna):
@@ -1752,10 +1754,10 @@ cdef class DatetimeValidator(TemporalValidator):
         return is_null_datetime64(value)
 
 
-cpdef bint is_datetime_array(ndarray values):
+cpdef bint is_datetime_array(ndarray values, bint skipna=True):
     cdef:
         DatetimeValidator validator = DatetimeValidator(len(values),
-                                                        skipna=True)
+                                                        skipna=skipna)
     return validator.validate(values)
 
 
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index fb266b4abba51..746fd140e48a1 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -4701,7 +4701,10 @@ def _maybe_promote(self, other: "Index"):
         """
 
         if self.inferred_type == "date" and isinstance(other, ABCDatetimeIndex):
-            return type(other)(self), other
+            try:
+                return type(other)(self), other
+            except OutOfBoundsDatetime:
+                return self, other
         elif self.inferred_type == "timedelta" and isinstance(other, ABCTimedeltaIndex):
             # TODO: we dont have tests that get here
             return type(other)(self), other
diff --git a/pandas/tests/dtypes/test_inference.py b/pandas/tests/dtypes/test_inference.py
index 4c4a5547247fc..e97716f7a5e9c 100644
--- a/pandas/tests/dtypes/test_inference.py
+++ b/pandas/tests/dtypes/test_inference.py
@@ -1106,6 +1106,21 @@ def test_date(self):
         result = lib.infer_dtype(dates, skipna=True)
         assert result == "date"
 
+    @pytest.mark.parametrize(
+        "values",
+        [
+            [date(2020, 1, 1), pd.Timestamp("2020-01-01")],
+            [pd.Timestamp("2020-01-01"), date(2020, 1, 1)],
+            [date(2020, 1, 1), pd.NaT],
+            [pd.NaT, date(2020, 1, 1)],
+        ],
+    )
+    @pytest.mark.parametrize("skipna", [True, False])
+    def test_infer_dtype_date_order_invariant(self, values, skipna):
+        # https://github.com/pandas-dev/pandas/issues/33741
+        result = lib.infer_dtype(values, skipna=skipna)
+        assert result == "date"
+
     def test_is_numeric_array(self):
 
         assert lib.is_float_array(np.array([1, 2.0]))
diff --git a/pandas/tests/indexes/datetimes/test_indexing.py b/pandas/tests/indexes/datetimes/test_indexing.py
index 097ee20534e4e..f08472fe72631 100644
--- a/pandas/tests/indexes/datetimes/test_indexing.py
+++ b/pandas/tests/indexes/datetimes/test_indexing.py
@@ -1,4 +1,4 @@
-from datetime import datetime, time, timedelta
+from datetime import date, datetime, time, timedelta
 
 import numpy as np
 import pytest
@@ -575,6 +575,38 @@ def test_get_indexer(self):
         with pytest.raises(ValueError, match="abbreviation w/o a number"):
             idx.get_indexer(idx[[0]], method="nearest", tolerance="foo")
 
+    @pytest.mark.parametrize(
+        "target",
+        [
+            [date(2020, 1, 1), pd.Timestamp("2020-01-02")],
+            [pd.Timestamp("2020-01-01"), date(2020, 1, 2)],
+        ],
+    )
+    def test_get_indexer_mixed_dtypes(self, target):
+        # https://github.com/pandas-dev/pandas/issues/33741
+        values = pd.DatetimeIndex(
+            [pd.Timestamp("2020-01-01"), pd.Timestamp("2020-01-02")]
+        )
+        result = values.get_indexer(target)
+        expected = np.array([0, 1], dtype=np.intp)
+        tm.assert_numpy_array_equal(result, expected)
+
+    @pytest.mark.parametrize(
+        "target, positions",
+        [
+            ([date(9999, 1, 1), pd.Timestamp("2020-01-01")], [-1, 0]),
+            ([pd.Timestamp("2020-01-01"), date(9999, 1, 1)], [0, -1]),
+            ([date(9999, 1, 1), date(9999, 1, 1)], [-1, -1]),
+        ],
+    )
+    def test_get_indexer_out_of_bounds_date(self, target, positions):
+        values = pd.DatetimeIndex(
+            [pd.Timestamp("2020-01-01"), pd.Timestamp("2020-01-02")]
+        )
+        result = values.get_indexer(target)
+        expected = np.array(positions, dtype=np.intp)
+        tm.assert_numpy_array_equal(result, expected)
+
 
 class TestMaybeCastSliceBound:
     def test_maybe_cast_slice_bounds_empty(self):