Mopar and jrpinball are right.

Score motors are cooler than most people realize. When AC power is applied to the score motor coil a changing magnetic field is created that is carried through the C shaped laminated frame to the round hole that surrounds the rotor. The changing magnetic field causes the rotor to turn which spins up the gear train and the cams.

Usually score motors need to stop quickly when the switch on the index cam opens so it starts and stops in the same position every time. When it's turning though the rotor has too much angular momentum to stop quickly enough. Even when the power is cut the rotor will coast to a stop rather than stopping suddenly which is essentially what you're seeing. The rotor is coasting enough to carry the motor forward enough to close the switch that fires it up again.

To get around this problem the rotor slides in and out of the laminated frame on its axis. When power is applied the magnetic field not only gets the rotor turning, it also pulls the rotor into the laminated frame so the gear on the end of the rotor's shaft engages with the rest of the gear train. When the power is cut the rotor keeps spinning, but a spring wound around its shaft and/or gravity are supposed to push the rotor out of the laminated frame which disengages the gear on the rotor's shaft from the rest of the gear train. The rotor can spin down at its own pace while the rest of the motor can stop much more quickly. It's basically an electromagnetic clutch.

Ordinarily a simple AC motor like this could start up in either direction but the two heavy copper bands wound around the laminated frame manipulate the magnetic field just enough that the rotor always spins up in the same direction. The whole thing is a simple, elegant design.

/Mark