what you can say, if a*a = e,and a is NOT the identity, is that a is of order 2, and that a is its own inverse (showing that elements of order 2 and of order 1(the identity is the only element of order 1) share this property).
a good example of an element of order 2 is the following group, which is a subset of the actions you can perform on a light switch:
{flip the switch, do nothing} and the group operation is: first do one thing, then the other.
"flip the switch" is an element of order 2: flip the switch, then flip the switch = do nothing.
(to be honest, this is in all actuality just Z2 in disguise, and is one of the basic ways in which logical circuits can "do math").