The reversing valve.
The part that makes a heat pump a heat pump. Stuck, it can heat your house in July.
A heat pump is an air conditioner that can run in either direction. The reversing valve is what makes that possible.
It is a four-port valve on the refrigerant circuit with a sliding mechanism inside, shifted by a small solenoid coil. In one position the indoor coil evaporates and the outdoor coil condenses, which is cooling. Shift it and the roles swap, which is heating.
It is also what performs the defrost cycle, briefly reversing to melt frost off the outdoor coil in winter.
How a stuck one behaves
Stuck in heating while you call for cooling. The system runs, the compressor runs, everything looks fine, and warm air comes out of the vents in July. People reasonably assume the whole system is broken when every component is working.
That is why the warm air guide says to rule out a reversing valve before anybody starts discussing refrigerant.
Stuck in cooling while you call for heat. The winter version, less dramatic in this climate but the same fault.
Stuck mid-position, which is the worst case: refrigerant bypasses internally, the system runs with poor capacity in both directions, and nothing looks obviously wrong on a gauge set at first glance.
Not shifting for defrost, which leaves the outdoor coil iced up all winter.
Valve or solenoid
Worth separating, because the prices are not comparable.
The solenoid coil is an electrical part on the outside of the valve body, and it is comparatively cheap. A coil that has failed or is not being energised looks exactly like a failed valve from the symptom.
The valve body is brazed into the refrigerant circuit. Replacing it means recovery, cutting, brazing, evacuation and recharge, which is a substantial job and routinely past the one thousand dollar ceiling here.
So if somebody has quoted a reversing valve, it is fair to ask whether the coil was tested first, and whether it was being energised at all. That is a measurement, and it is a large difference in price.