It's effectively an electric motor - possibly an escapement release solenoid - but probably it provides he complete driving power for the flip action. If here is no other timing or driving mechanism then it may have been run as a "slave" with control pulses sent via the visible wiring from a central controller.

In his photo from your website you can see a general view of how it is associated with the shaft drive.

![enter image description here][1]


And here you can see the reduction gear train from the motor shaft to the flipping shaft.  

![enter image description here][2]


  [1]: https://i.sstatic.net/g7D95.jpg
  [2]: https://i.sstatic.net/mEB40.jpg