Originally posted by: Paperdoc
At the level of basic theory OP is on the right track. A heat pump fundamentally is used to "pump" heat from a low-temperature zone to a higher-temperature zone, the exact opposite of what normally happens for heat flow. In doing so it consumes outside energy (electrical) to power the compressor which is converted to heat that is exhausted to the hot side of the system. So from the perspective of the hot side, the heat available is the electrical energy consumed PLUS the heat moved from cold to hot side, and the result is that the hot side receives MORE heat than just the electrical heat. Thus it appears to be MORE than 100% "efficient" in converting electrical energy to heat energy. In that sense it actually provides more heat to the room than an electrical heater of the same power consumption. Now, nothing is more than 100% efficient - the point to recognize is that the extra heat does not just magically appear - it is moved from the cold side to hot side.
AC efficiency really is related to heat transfer rate divided by electrical energy use rate - that is, BTU/hr divided by Watts consumed. As you might expect, this is highest when the temperature difference between "hot" and "cold" is small, so the compressor needs to "push" heat up a small step to the "hot" side. The larger the difference, the lower the efficiency of the AC. So even if it worked, efficiency (that is, really, how much "free" heat you pump on top of how much is just from electrical energy that runs the motor) would not be very good in the winter when the indoor "hot" side may be 30C (55F) degrees higher than the outdoor "cold" side.
However, the devil is in the details. There are many versions of heat pumps. They all differ at the core in what temperature ranges are the "cold" and "hot" sides operating at? The type of refrigerant and the pressures around the circuit are based on those answers. Then that leads to the design of the compressor, the two heat exchangers (the Evaporator and the Condenser) and their piping, and the control systems and their setpoints. A through-the-window AC is designed to operate with a "hot" side of from 0 to 50C (32 to 122F), approximately, and a "cold" side in a similar temperature range. NEITHER side can operate below 0C (32 F), both because of the refrigerant characteristics and because you can't allow the heat exchanger fins on either side to get plugged up with water frost. We also should recognize that the air temperature on the "hot" side must be LOWER than that of the compressed refrigerant circulating through the Condenser; otherwise there would be no heat flow into the "hot" air and no condensing action.
Bottom line, mounting a window AC backwards in the window and creating a thermostat system to work "backwards", too (so it turns on when the "Hot" side is too cool, rather than when the "cold" side is too warm) might only work over a limited range of temperatures. And it might break itself, because it is not designed to prevent it from operating under the wrong conditions that could be created that way. It was only designed to anticipate and protect itself from other conditions expected when it is used for its original purpose.