diff --git a/qiskit/transpiler/passes/utils/gate_direction.py b/qiskit/transpiler/passes/utils/gate_direction.py
index 323f19bf7fe9..262538a2fad0 100644
--- a/qiskit/transpiler/passes/utils/gate_direction.py
+++ b/qiskit/transpiler/passes/utils/gate_direction.py
@@ -19,7 +19,7 @@
 
 from qiskit.converters import dag_to_circuit, circuit_to_dag
 from qiskit.circuit import QuantumRegister, ControlFlowOp
-from qiskit.dagcircuit import DAGCircuit
+from qiskit.dagcircuit import DAGCircuit, DAGOpNode
 from qiskit.circuit.library.standard_gates import (
     RYGate,
     HGate,
@@ -34,6 +34,10 @@
 )
 
 
+def _swap_node_qargs(node):
+    return DAGOpNode(node.op, node.qargs[::-1], node.cargs)
+
+
 class GateDirection(TransformationPass):
     """Modify asymmetric gates to match the hardware coupling direction.
 
@@ -58,6 +62,8 @@ class GateDirection(TransformationPass):
              └──────┘          └───┘└──────┘└───┘
     """
 
+    _KNOWN_REPLACEMENTS = frozenset(["cx", "cz", "ecr", "swap", "rzx", "rxx", "ryy", "rzz"])
+
     def __init__(self, coupling_map, target=None):
         """GateDirection pass.
 
@@ -99,6 +105,8 @@ def __init__(self, coupling_map, target=None):
         self._swap_dag.add_qreg(qr)
         self._swap_dag.apply_operation_back(SwapGate(), [qr[1], qr[0]], [])
 
+        # If adding more replacements (either static or dynamic), also update the class variable
+        # `_KNOWN_REPLACMENTS` to include them in the error messages.
         self._static_replacements = {
             "cx": self._cx_dag,
             "cz": self._cz_dag,
@@ -187,8 +195,9 @@ def _run_coupling_map(self, dag, wire_map, edges=None):
                     dag.substitute_node_with_dag(node, self._rzz_dag(*node.op.params))
                 else:
                     raise TranspilerError(
-                        f"Flipping of gate direction is only supported "
-                        f"for {list(self._static_replacements)} at this time, not '{node.name}'."
+                        f"'{node.name}' would be supported on '{qargs}' if the direction were"
+                        f" swapped, but no rules are known to do that."
+                        f" {list(self._KNOWN_REPLACEMENTS)} can be automatically flipped."
                     )
         return dag
 
@@ -281,11 +290,22 @@ def _run_target(self, dag, wire_map):
                         f"The circuit requires a connection between physical qubits {qargs}"
                         f" for {node.name}"
                     )
+            elif self.target.instruction_supported(node.name, qargs):
+                continue
+            elif self.target.instruction_supported(node.name, swapped) or dag.has_calibration_for(
+                _swap_node_qargs(node)
+            ):
+                raise TranspilerError(
+                    f"'{node.name}' would be supported on '{qargs}' if the direction were"
+                    f" swapped, but no rules are known to do that."
+                    f" {list(self._KNOWN_REPLACEMENTS)} can be automatically flipped."
+                )
             else:
                 raise TranspilerError(
-                    f"Flipping of gate direction is only supported "
-                    f"for {list(self._static_replacements)} at this time, not '{node.name}'."
+                    f"'{node.name}' with parameters '{node.op.params}' is not supported on qubits"
+                    f" '{qargs}' in either direction."
                 )
+
         return dag
 
     def run(self, dag):
diff --git a/test/python/transpiler/test_gate_direction.py b/test/python/transpiler/test_gate_direction.py
index 06c1d814b147..a45568efdf58 100644
--- a/test/python/transpiler/test_gate_direction.py
+++ b/test/python/transpiler/test_gate_direction.py
@@ -439,6 +439,49 @@ def test_target_control_flow(self):
         pass_ = GateDirection(None, target)
         self.assertEqual(pass_(circuit), expected)
 
+    def test_target_cannot_flip_message(self):
+        """A suitable error message should be emitted if the gate would be supported if it were
+        flipped."""
+        gate = Gate("my_2q_gate", 2, [])
+        target = Target(num_qubits=2)
+        target.add_instruction(gate, properties={(0, 1): None})
+
+        circuit = QuantumCircuit(2)
+        circuit.append(gate, (1, 0))
+
+        pass_ = GateDirection(None, target)
+        with self.assertRaisesRegex(TranspilerError, "'my_2q_gate' would be supported.*"):
+            pass_(circuit)
+
+    def test_target_cannot_flip_message_calibrated(self):
+        """A suitable error message should be emitted if the gate would be supported if it were
+        flipped."""
+        target = Target(num_qubits=2)
+        target.add_instruction(CXGate(), properties={(0, 1): None})
+
+        gate = Gate("my_2q_gate", 2, [])
+        circuit = QuantumCircuit(2)
+        circuit.append(gate, (1, 0))
+        circuit.add_calibration(gate, (0, 1), pulse.ScheduleBlock())
+
+        pass_ = GateDirection(None, target)
+        with self.assertRaisesRegex(TranspilerError, "'my_2q_gate' would be supported.*"):
+            pass_(circuit)
+
+    def test_target_unknown_gate_message(self):
+        """A suitable error message should be emitted if the gate isn't valid in either direction on
+        the target."""
+        gate = Gate("my_2q_gate", 2, [])
+        target = Target(num_qubits=2)
+        target.add_instruction(CXGate(), properties={(0, 1): None})
+
+        circuit = QuantumCircuit(2)
+        circuit.append(gate, (0, 1))
+
+        pass_ = GateDirection(None, target)
+        with self.assertRaisesRegex(TranspilerError, "'my_2q_gate'.*not supported on qubits .*"):
+            pass_(circuit)
+
     def test_allows_calibrated_gates_coupling_map(self):
         """Test that the gate direction pass allows a gate that's got a calibration to pass through
         without error."""