I don't know if this is correct, but the head and blades can turn without turning the motor. Inversely, when the motor starts, it doesn't turn the head immediately, but only eventually, due to friction. I'm pretty sure when we first got the CP2, turning the head actually turned the big gear and therefore the motor.
Looking at the head up close, I see there are two pins that are sticking out.
The plastic assembly where the pins are sticking out of rotates freely against the main shaft.
Solution: The Jesus pin is broken. We've tried a sewing needle, but they're dangerous to cut (the pieces fly across the room when the pliers get pinch it through). Now we're using a paper clip, with the ends bent up because the hole in the plastic has widened sufficiently that friction is not enough to hold the clip in place. Paper clips actually break even easier because the metal is very soft, but I think this might prolong the life of other components.