diff --git a/.github/workflows/ci.yaml b/.github/workflows/ci.yaml
index ba692e4..40abd8c 100644
--- a/.github/workflows/ci.yaml
+++ b/.github/workflows/ci.yaml
@@ -65,6 +65,10 @@ jobs:
             python=${{matrix.python-version}}
             conda
 
+      - name: Install nightly xarray
+        run: |
+          python -m pip install --upgrade --pre -i https://pypi.anaconda.org/scientific-python-nightly-wheels/simple xarray
+
       - name: Install xarray-array-testing
         run: |
           python -m pip install --no-deps -e .
diff --git a/xarray_array_testing/reduction.py b/xarray_array_testing/reduction.py
index fb4b158..38b37e9 100644
--- a/xarray_array_testing/reduction.py
+++ b/xarray_array_testing/reduction.py
@@ -1,6 +1,7 @@
 from contextlib import nullcontext
 
 import hypothesis.strategies as st
+import numpy as np
 import pytest
 import xarray.testing.strategies as xrst
 from hypothesis import given
@@ -24,4 +25,76 @@ def test_variable_numerical_reduce(self, op, data):
             # compute using xp.<OP>(array)
             expected = getattr(self.xp, op)(variable.data)
 
+        assert isinstance(actual, self.array_type), f"wrong type: {type(actual)}"
+        self.assert_equal(actual, expected)
+
+    @pytest.mark.parametrize("op", ["all", "any"])
+    @given(st.data())
+    def test_variable_boolean_reduce(self, op, data):
+        variable = data.draw(xrst.variables(array_strategy_fn=self.array_strategy_fn))
+
+        with self.expected_errors(op, variable=variable):
+            # compute using xr.Variable.<OP>()
+            actual = getattr(variable, op)().data
+            # compute using xp.<OP>(array)
+            expected = getattr(self.xp, op)(variable.data)
+
+        assert isinstance(actual, self.array_type), f"wrong type: {type(actual)}"
+        self.assert_equal(actual, expected)
+
+    @pytest.mark.parametrize("op", ["max", "min"])
+    @given(st.data())
+    def test_variable_order_reduce(self, op, data):
+        variable = data.draw(xrst.variables(array_strategy_fn=self.array_strategy_fn))
+
+        with self.expected_errors(op, variable=variable):
+            # compute using xr.Variable.<OP>()
+            actual = getattr(variable, op)().data
+            # compute using xp.<OP>(array)
+            expected = getattr(self.xp, op)(variable.data)
+
+        assert isinstance(actual, self.array_type), f"wrong type: {type(actual)}"
+        self.assert_equal(actual, expected)
+
+    @pytest.mark.parametrize("op", ["argmax", "argmin"])
+    @given(st.data())
+    def test_variable_order_reduce_index(self, op, data):
+        variable = data.draw(xrst.variables(array_strategy_fn=self.array_strategy_fn))
+
+        with self.expected_errors(op, variable=variable):
+            # compute using xr.Variable.<OP>()
+            actual = {k: v.item() for k, v in getattr(variable, op)(dim=...).items()}
+
+            # compute using xp.<OP>(array)
+            index = getattr(self.xp, op)(variable.data)
+            unraveled = np.unravel_index(index, variable.shape)
+            expected = dict(zip(variable.dims, unraveled))
+
+        self.assert_equal(actual, expected)
+
+    @pytest.mark.parametrize(
+        "op",
+        [
+            "cumsum",
+            pytest.param(
+                "cumprod",
+                marks=pytest.mark.skip(reason="not yet included in the array api"),
+            ),
+        ],
+    )
+    @given(st.data())
+    def test_variable_cumulative_reduce(self, op, data):
+        array_api_names = {"cumsum": "cumulative_sum", "cumprod": "cumulative_prod"}
+        variable = data.draw(xrst.variables(array_strategy_fn=self.array_strategy_fn))
+
+        with self.expected_errors(op, variable=variable):
+            # compute using xr.Variable.<OP>()
+            actual = getattr(variable, op)().data
+            # compute using xp.<OP>(array)
+            # Variable implements n-d cumulative ops by iterating over dims
+            expected = variable.data
+            for axis in range(variable.ndim):
+                expected = getattr(self.xp, array_api_names[op])(expected, axis=axis)
+
+        assert isinstance(actual, self.array_type), f"wrong type: {type(actual)}"
         self.assert_equal(actual, expected)