Single-phase induction motors will happily run equally well in either direction, but they are not self-starting unless there is a auxiliary winding provided for that purpose.
What you have is almost certainly a "split-phase" induction motor, which has two windings. One winding is connected directly to the power source, and the other is connected in series with a capacitor. The capacitor provides phase shift necessary to determine the direction the motor will start turning when power is applied. If both windings in the motor are the same size the motor will be reversible, depending upon which winding the capacitor is connected to. If the windings are not the same size, the capacitor can only be connected to the smaller "auxiliary" winding.
If the windings of your motor still have continuity (i.e. not an open circuit), then it is indeed possible that a new capacitor will restore it to service. Capacitors are a lot cheaper than motors, so it's probably worth a try. Be sure you get a proper-value "run" capacitor, which will be rated for continuous duty, not a "start" capacitor, which is only intended for intermittent service.
[This message has been edited by Marvin McInnis (edited 04-03-2011).]
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