I did the same with a split phase motor, electrically the same as a cap start motor; it has starting and running windings but no capacitor.
But, there are also capacitor run motors, for higher torque, that have no centrifugal switch. The starting winding is energized continuously. They are bult for this duty. You should not try to run a starting winding continuously.
The drum switch I used had a wiring diagram, which I may still have. I'll look for it.
The idea is to always have the running winding on when running, and the starting winding plus capacitor, not just the capacitor, are reversed. The cap is made to run on AC and does not care about polarity.
You also need a motor where the winding leads are brought individually to the motor wiring box.
Alec Ryals wrote:
I can't find how to wire a 120V cap start motor for FWD & Rev using a
Reversing your motor is as simple as reversing the polarity of the motor
capacitor but without the wiring diagrams of your motor and switch it's
impossible to advice farther.
Ken from Canada.