Run electric motor CW & CCW with DP-DT switch
As the motor is capacitor start, the answer may be yes, if the four wires give you access to the main and start windings.

Try starting the motor with the start winding reversed, it may start in the opposite direction. The main winding stays connected as normal.

As the circuit diagram tool isn't saving for me at the moment, let's see if ASCIIart will do.

         DPDT
    cap     ______________________|
-----||---/                       |
            ----|                 |
            ____|--start winding--|
----------/                       |
            ----------------------|


To identify if you do have all the wires for all the windings, measure the resistances between the wires with a meter. You are looking for two isolated windings. The main one will be lower resistance.