There are two different types of motors. The two wire motor works exactly as you described, with the PTT relays serving to reverse the polarity of the voltage on the PTT motor. Power is applied to one of the PTT motor wires via the relay. The other wire on the PTT motor is grounded, when the relay is not energized.
Older PTT systems have a 3 wire motor. The motor is wound with a center winding ground. Power is applied to one side of the winding to spin the motor in the up direction. Alternatively, power is applied to the other side of the winding to spin the motor in the down direction. Solenoids provide power to one of the windings at a time. The other winding is left open.