I'm looking for hinge that will allow the gate on my wooden fence to swing both ways.

Thank you for writing. Unfortunately, that is easier said than done. The trouble with a wooden gate swinging both ways is that the wood will likely bind on one side of the other of the swing. A standard J-bolt hinge may be your best bet. Please read the following information which may also be helpful:
http://www.hooverfence.com/woodfence/ag-inst3.htm