diff --git a/cirq-core/cirq/transformers/gauge_compiling/gauge_compiling.py b/cirq-core/cirq/transformers/gauge_compiling/gauge_compiling.py
index 4341e82bc10..17c3c620a08 100644
--- a/cirq-core/cirq/transformers/gauge_compiling/gauge_compiling.py
+++ b/cirq-core/cirq/transformers/gauge_compiling/gauge_compiling.py
@@ -71,6 +71,7 @@ class ConstantGauge(Gauge):
     post_q1: Tuple[ops.Gate, ...] = field(
         default=(), converter=lambda g: (g,) if isinstance(g, ops.Gate) else tuple(g)
     )
+    swap_qubits: bool = False
 
     def sample(self, gate: ops.Gate, prng: np.random.Generator) -> "ConstantGauge":
         return self
@@ -85,6 +86,41 @@ def post(self) -> Tuple[Tuple[ops.Gate, ...], Tuple[ops.Gate, ...]]:
         """A tuple (ops to apply to q0, ops to apply to q1)."""
         return self.post_q0, self.post_q1
 
+    def on(self, q0: ops.Qid, q1: ops.Qid) -> ops.Operation:
+        """Returns the operation that replaces the two qubit gate."""
+        if self.swap_qubits:
+            return self.two_qubit_gate(q1, q0)
+        return self.two_qubit_gate(q0, q1)
+
+
+@frozen
+class SameGateGauge(Gauge):
+    """Same as ConstantGauge but the new two-qubit gate equals the old gate."""
+
+    pre_q0: Tuple[ops.Gate, ...] = field(
+        default=(), converter=lambda g: (g,) if isinstance(g, ops.Gate) else tuple(g)
+    )
+    pre_q1: Tuple[ops.Gate, ...] = field(
+        default=(), converter=lambda g: (g,) if isinstance(g, ops.Gate) else tuple(g)
+    )
+    post_q0: Tuple[ops.Gate, ...] = field(
+        default=(), converter=lambda g: (g,) if isinstance(g, ops.Gate) else tuple(g)
+    )
+    post_q1: Tuple[ops.Gate, ...] = field(
+        default=(), converter=lambda g: (g,) if isinstance(g, ops.Gate) else tuple(g)
+    )
+    swap_qubits: bool = False
+
+    def sample(self, gate: ops.Gate, prng: np.random.Generator) -> ConstantGauge:
+        return ConstantGauge(
+            two_qubit_gate=gate,
+            pre_q0=self.pre_q0,
+            pre_q1=self.pre_q1,
+            post_q0=self.post_q0,
+            post_q1=self.post_q1,
+            swap_qubits=self.swap_qubits,
+        )
+
 
 def _select(choices: Sequence[Gauge], probabilites: np.ndarray, prng: np.random.Generator) -> Gauge:
     return choices[prng.choice(len(choices), p=probabilites)]
@@ -154,7 +190,7 @@ def __call__(
                     gauge = self.gauge_selector(rng).sample(op.gate, rng)
                     q0, q1 = op.qubits
                     left.extend([g(q) for g in gs] for q, gs in zip(op.qubits, gauge.pre))
-                    center.append(gauge.two_qubit_gate(q0, q1))
+                    center.append(gauge.on(q0, q1))
                     right.extend([g(q) for g in gs] for q, gs in zip(op.qubits, gauge.post))
                 else:
                     center.append(op)
diff --git a/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge.py b/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge.py
index 96643eb3a90..88713ebba80 100644
--- a/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge.py
+++ b/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge.py
@@ -17,17 +17,17 @@
 from cirq.transformers.gauge_compiling.gauge_compiling import (
     GaugeTransformer,
     GaugeSelector,
-    ConstantGauge,
+    SameGateGauge,
 )
 from cirq import ops
 
 SpinInversionGaugeSelector = GaugeSelector(
     gauges=[
-        ConstantGauge(two_qubit_gate=ops.ZZ, pre_q0=ops.X, post_q0=ops.X),
-        ConstantGauge(two_qubit_gate=ops.ZZ, pre_q1=ops.X, post_q1=ops.X),
+        SameGateGauge(pre_q0=ops.X, post_q0=ops.X, pre_q1=ops.X, post_q1=ops.X),
+        SameGateGauge(),
     ]
 )
 
 SpinInversionGaugeTransformer = GaugeTransformer(
-    target=ops.ZZ, gauge_selector=SpinInversionGaugeSelector
+    target=ops.GateFamily(ops.ZZPowGate), gauge_selector=SpinInversionGaugeSelector
 )
diff --git a/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge_test.py b/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge_test.py
index 6e38ff451f1..c599b328316 100644
--- a/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge_test.py
+++ b/cirq-core/cirq/transformers/gauge_compiling/spin_inversion_gauge_test.py
@@ -12,12 +12,26 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-
 import cirq
 from cirq.transformers.gauge_compiling import SpinInversionGaugeTransformer
 from cirq.transformers.gauge_compiling.gauge_compiling_test_utils import GaugeTester
 
 
-class TestSpinInversionGauge(GaugeTester):
+class TestSpinInversionGauge_0(GaugeTester):
     two_qubit_gate = cirq.ZZ
     gauge_transformer = SpinInversionGaugeTransformer
+
+
+class TestSpinInversionGauge_1(GaugeTester):
+    two_qubit_gate = cirq.ZZ**0.1
+    gauge_transformer = SpinInversionGaugeTransformer
+
+
+class TestSpinInversionGauge_2(GaugeTester):
+    two_qubit_gate = cirq.ZZ**-1
+    gauge_transformer = SpinInversionGaugeTransformer
+
+
+class TestSpinInversionGauge_3(GaugeTester):
+    two_qubit_gate = cirq.ZZ**0.3
+    gauge_transformer = SpinInversionGaugeTransformer
diff --git a/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge.py b/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge.py
index 2e6e96f9220..60252d6ad56 100644
--- a/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge.py
+++ b/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge.py
@@ -12,18 +12,53 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-"""A Gauge transformer for CZ**0.5 gate."""
+"""A Gauge transformer for CZ**0.5 and CZ**-0.5 gates."""
+
+
+from typing import TYPE_CHECKING
+import numpy as np
 
 from cirq.transformers.gauge_compiling.gauge_compiling import (
     GaugeTransformer,
     GaugeSelector,
     ConstantGauge,
+    Gauge,
 )
-from cirq.ops.common_gates import CZ
-from cirq import ops
+from cirq.ops import CZ, S, X, Gateset
+
+if TYPE_CHECKING:
+    import cirq
+
+_SQRT_CZ = CZ**0.5
+_ADJ_S = S**-1
 
-SqrtCZGaugeSelector = GaugeSelector(
-    gauges=[ConstantGauge(pre_q0=ops.X, post_q0=ops.X, post_q1=ops.Z**0.5, two_qubit_gate=CZ**-0.5)]
-)
 
-SqrtCZGaugeTransformer = GaugeTransformer(target=CZ**0.5, gauge_selector=SqrtCZGaugeSelector)
+class SqrtCZGauge(Gauge):
+
+    def weight(self) -> float:
+        return 3.0
+
+    def sample(self, gate: 'cirq.Gate', prng: np.random.Generator) -> ConstantGauge:
+        if prng.choice([True, False]):
+            return ConstantGauge(two_qubit_gate=gate)
+        swap_qubits = prng.choice([True, False])
+        if swap_qubits:
+            return ConstantGauge(
+                pre_q1=X,
+                post_q1=X,
+                post_q0=S if gate == _SQRT_CZ else _ADJ_S,
+                two_qubit_gate=gate**-1,
+                swap_qubits=True,
+            )
+        else:
+            return ConstantGauge(
+                pre_q0=X,
+                post_q0=X,
+                post_q1=S if gate == _SQRT_CZ else _ADJ_S,
+                two_qubit_gate=gate**-1,
+            )
+
+
+SqrtCZGaugeTransformer = GaugeTransformer(
+    target=Gateset(_SQRT_CZ, _SQRT_CZ**-1), gauge_selector=GaugeSelector(gauges=[SqrtCZGauge()])
+)
diff --git a/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge_test.py b/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge_test.py
index 8d18c208e1d..e6e871cf938 100644
--- a/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge_test.py
+++ b/cirq-core/cirq/transformers/gauge_compiling/sqrt_cz_gauge_test.py
@@ -12,7 +12,6 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
-
 import cirq
 from cirq.transformers.gauge_compiling import SqrtCZGaugeTransformer
 from cirq.transformers.gauge_compiling.gauge_compiling_test_utils import GaugeTester
@@ -21,3 +20,8 @@
 class TestSqrtCZGauge(GaugeTester):
     two_qubit_gate = cirq.CZ**0.5
     gauge_transformer = SqrtCZGaugeTransformer
+
+
+class TestAdjointSqrtCZGauge(GaugeTester):
+    two_qubit_gate = cirq.CZ**-0.5
+    gauge_transformer = SqrtCZGaugeTransformer