Lathe Reverse

[m_c]

There are 3 variations of common induction single phase motors.

The fact you've only got one capacitor rules out the Capacitor Start / Capacitor Run option (provides the best of both options below)
You say you don't hear a click, which likely rules it out being a Capacitor Start motor (gives more starting torque, but adds the complexity of a centrifugal switch)
So it's likely you have a Capacitor Run motor (gives very little starting torque, but helps the motor run a bit smoother)

They ultimately all work in a similar way, with a second (usually weaker) winding, that uses a capacitor to create a delayed phase, otherwise the motor will simply vibrate between poles when you apply AC power.
The bigger the capacitor, the more power the second winding gets (more starting torque), but the more the windings will heat up due to the relationship between the power, the motors inductions causing the windings to essentially fight against each other (hence why a Cap Start motor relies on a switch to disconnect once running, otherwise the motor would overheat quite quickly).

You just need to move the capacitor's position so it creates the delayed phase in the correct direction, but I can never remember the connections.