I don't know the true philosophical reason, but $Z_2$ symmetry is really
omnipresent in Matematics and in the nature. For example, most animals, including practically
all vertebrate animals
(like ourselves) have aproximately $Z_2$ symmetric bodies, and no
larger group. This suggests that $Z_2$ was the favorite group of the Creator, at least in that
period of his activity when we was creating advanced animals:-)

If you prefer Evolution, this $Z_2$ symmetry must somehow be explained by the survival
of the fittest. I don't know exactly how, but this suggests that this is a very important group.
Notice that plants, mushrooms, and simplest animals usually do not have it.

As a result of this (2-fold symmetry of animal bodies) we tend to like this kind of symmetry.
Look at all our technology: cars, ships, airplanes, etc. They all have 2-fold symmetry,
at least from outside (like our bodies, they also have this symmetry only outside).
Once my friend, an airspace engineer, told me that there was a project of an airplane which did not
have this outside 2-fold symmetry. The project was rejected for the only reason that
"no one will want to fly in such an airplane". I am serious:
http://en.wikipedia.org/wiki/Oblique_wing

In mathematics, from my personal perspective, it is $z\mapsto\overline{z}$ first of all.
(Once I even proposed
to my co-author to call one of our papers "Some applications of representation theory of $Z_2$");
the paper was full of different representations of this group,
We were working on real algebraic geometry.)

This very same symmetry $z\mapsto\overline{z}$
is also hidden in Hermitian symmetry, $C^*$ algebras, all sorts of "duality" everywhere, etc.
Which suggests that the Creator
of the Universe always had a strong bias in favor of this particular group.

reversibleelements in groups, occurred very naturally across mathematics. Unfortunately I don't have references to hand but he might have more on his webpage – Yemon Choi Apr 4 '13 at 22:41