The reversing valve decides which coil is hot and which is cold. A small solenoid coil on the outside moves a pilot, and refrigerant pressure then slides the piston inside the valve body. The question is always whether the electrical side failed or the valve itself.
A dead solenoid coil, or a thermostat signal that never reaches it, is the cheaper case: a new coil goes on without opening the refrigerant circuit. A valve body that is stuck, or moves only partway and leaks hot gas to the suction side, has to be cut out, with the refrigerant recovered, a new valve brazed in, a vacuum pulled and the system recharged. The partial shift is the sneaky one, because the system still heats and cools, just weakly in both seasons.
Typical Bay County market range for a reversing valve, refrigerant recovered and recharged: $500 to $1,400 (not a Quincy’s quote).