diff --git a/cirq-core/cirq/circuits/quil_output_test.py b/cirq-core/cirq/circuits/quil_output_test.py
index d67094f6624..28986768be8 100644
--- a/cirq-core/cirq/circuits/quil_output_test.py
+++ b/cirq-core/cirq/circuits/quil_output_test.py
@@ -233,6 +233,7 @@ def test_all_operations():
 CPHASE({np.pi / 2}) 0 1
 RY({np.pi / 2}) 1
 SWAP 0 1
+SWAP 1 0
 PSWAP({3 * np.pi / 4}) 0 1
 H 2
 CCNOT 0 1 2
@@ -324,6 +325,7 @@ def _all_operations(q0, q1, q2, q3, q4, include_measurements=True):
         cirq.CNOT(q0, q1),
         cirq.CNOT(q0, q1) ** 0.5,  # Requires 2-qubit decomposition
         cirq.SWAP(q0, q1),
+        cirq.SWAP(q1, q0) ** -1,
         cirq.SWAP(q0, q1) ** 0.75,  # Requires 2-qubit decomposition
         cirq.CCZ(q0, q1, q2),
         cirq.CCX(q0, q1, q2),
diff --git a/cirq-core/cirq/ops/swap_gates.py b/cirq-core/cirq/ops/swap_gates.py
index 72b9a1c41ba..5d5fa0174d4 100644
--- a/cirq-core/cirq/ops/swap_gates.py
+++ b/cirq-core/cirq/ops/swap_gates.py
@@ -143,7 +143,7 @@ def _qasm_(self, args: 'cirq.QasmArgs', qubits: Tuple['cirq.Qid', ...]) -> Optio
     def _quil_(
         self, qubits: Tuple['cirq.Qid', ...], formatter: 'cirq.QuilFormatter'
     ) -> Optional[str]:
-        if self._exponent == 1:
+        if self._exponent % 2 == 1:
             return formatter.format('SWAP {0} {1}\n', qubits[0], qubits[1])
         return formatter.format(
             'PSWAP({0}) {1} {2}\n', self._exponent * np.pi, qubits[0], qubits[1]