So why not just leave the starter winding connected? The centrifugal switch S is mounted to the back plate and is activated by a disk P that pushes against a tab on the switch just left of the S in the photo. If both become energized, some form of mechanical interlock is required to prevent both armatures from pulling in.
Basically, we need to swap the connections on either end of the starter winding. “Single Phase Ac Motor Reversing Switch Wiring Diagram”
