Some are reversible, synchronous, may or may not require a capacitor. Sometimes the capacitor is just used to start the motor, other times it stays in all the time. I have a lot to learn about AC motors, so don't expect to get the last word from me -- yet.

The best way to work with a reversible motor is with something called a drum switch. This has 3 positions (forward, stop, and reverse). Such a switch typically has 6 connections and can be tricky to wire up. Here is a nice datasheet from Square D with lots of wiring diagrams for different kinds of motors.

A fellow named Grant Erwin wrote a nice essay on AC motors, with particular reference to the home machine shop. I have a local copy right here since I liked this so much and was worried it might go away.

Here is a nice manual on Berger Lahr RSM series motors (Reversible Synchronous Motors), which is quite relevant since that happens to be the very motor I am now struggling with.