diff --git a/cf_xarray/accessor.py b/cf_xarray/accessor.py
index 35692bfc..5be0efbd 100644
--- a/cf_xarray/accessor.py
+++ b/cf_xarray/accessor.py
@@ -1261,7 +1261,9 @@ def _maybe_to_dataarray(self, obj=None):
             return obj
 
     def rename_like(
-        self, other: Union[DataArray, Dataset]
+        self,
+        other: Union[DataArray, Dataset],
+        skip: Union[str, Iterable[str]] = None,
     ) -> Union[DataArray, Dataset]:
         """
         Renames variables in object to match names of like-variables in ``other``.
@@ -1277,20 +1279,30 @@ def rename_like(
         ----------
         other : DataArray, Dataset
             Variables will be renamed to match variable names in this xarray object
+        skip: str, Iterable[str], optional
+            Limit the renaming excluding
+            ("axes", "cell_measures", "coordinates", "standard_names")
+            or a subset thereof.
 
         Returns
         -------
         DataArray or Dataset with renamed variables
         """
+        skip = [skip] if isinstance(skip, str) else skip or []
+
         ourkeys = self.keys()
         theirkeys = other.cf.keys()
 
         good_keys = ourkeys & theirkeys
         keydict = {}
         for key in good_keys:
-            ours = _get_all(self._obj, key)
-            theirs = _get_all(other, key)
-            keydict[key] = dict(ours=ours, theirs=theirs)
+            ours = set(_get_all(self._obj, key))
+            theirs = set(_get_all(other, key))
+            for attr in skip:
+                ours -= set(getattr(self, attr).get(key, []))
+                theirs -= set(getattr(other.cf, attr).get(key, []))
+            if ours and theirs:
+                keydict[key] = dict(ours=list(ours), theirs=list(theirs))
 
         conflicts = {}
         for k0, v0 in keydict.items():
@@ -1299,7 +1311,7 @@ def rename_like(
                 continue
             for v1 in keydict.values():
                 # Conflicts have same ours but different theirs or vice versa
-                if sum([v0["ours"] == v1["ours"], v0["theirs"] == v1["theirs"]]) == 1:
+                if (v0["ours"] == v1["ours"]) != (v0["theirs"] == v1["theirs"]):
                     conflicts[k0] = v0
                     break
         if conflicts:
diff --git a/cf_xarray/tests/test_accessor.py b/cf_xarray/tests/test_accessor.py
index 71d35441..bab850a9 100644
--- a/cf_xarray/tests/test_accessor.py
+++ b/cf_xarray/tests/test_accessor.py
@@ -250,8 +250,14 @@ def test_rename_like():
     assert "TEMP" in renamed
 
     # skip conflicting variables
+    da = popds.cf["TEMP"]
     with pytest.warns(UserWarning, match="Conflicting variables skipped:.*"):
-        popds.cf.rename_like(airds)
+        expected = {"longitude": ["TLONG"], "latitude": ["TLAT"]}
+        actual = da.cf.rename_like(airds).cf.coordinates
+        assert expected == actual
+    expected = {"longitude": ["lon"], "latitude": ["lat"]}
+    actual = da.cf.rename_like(airds, skip="axes").cf.coordinates
+    assert expected == actual
 
 
 @pytest.mark.parametrize("obj", objects)
diff --git a/doc/examples/introduction.ipynb b/doc/examples/introduction.ipynb
index 290444f8..1a2a38ef 100644
--- a/doc/examples/introduction.ipynb
+++ b/doc/examples/introduction.ipynb
@@ -605,10 +605,10 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "If we drop the `X` and `Y` axes, a one-to-one mapping is possible. In this\n",
-    "example, `TLONG` and `TLAT` are renamed to `lon` and `lat` i.e. their\n",
-    "counterparts in `ds`. Note the the `coordinates` attribute is appropriately\n",
-    "changed.\n"
+    "If we exclude all axes (variables with `axis` attribute), a one-to-one mapping\n",
+    "is possible. In this example, `TLONG` and `TLAT` are renamed to `lon` and `lat`\n",
+    "i.e. their counterparts in `ds`. Note the the `coordinates` attribute is\n",
+    "appropriately changed.\n"
    ]
   },
   {
@@ -622,8 +622,7 @@
    },
    "outputs": [],
    "source": [
-    "da = da.cf.drop_vars([\"X\", \"Y\"])\n",
-    "da.cf.rename_like(ds)"
+    "da.cf.rename_like(ds, skip=\"axes\")"
    ]
   },
   {