<a>=<a^-1> in all cases.

Why ? Well <a^-1> contains a (the group generated by an element contains the inverse of this element). So we have a=(a^-1)^k for some k in N (k less or equal to the cardinal of the group). So a power of a is also a power of a^-1, because a^n=(a^-1)^kn.

And vice versa for the other inclusion.