Is there a deep reason for the fecundity of involutions? You might have come across the book of involutions in your travels. A colleague of mine asked whether there is a natural global reason (versus ad-hoc trickery) for considering involutions in mathematics. The above book provides many situations that suggest such a global perspective. Having witnessed the extraordinary power of certain involutions in operator algebras (e.g. in Tomita-Takesaki theory), I'd be interested in hearing about such a global perspective in summary from an expert. I'm aware that this question as I've asked it risks being trite...perhaps warranting the answer "it's the simplest nontrivial symmetry" but the existence of the above book might suggest otherwise:

Question: What are some "global" reasons for considering involutions in mathematics?

 A: 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.
A: It all boils down to the fact that $\mathbb{R}$ has two ends!
In all (most) mathematical processes there is some notion of direction: counting, moving (along a curve), mapping from one space into another, reading a formula from left to right... the geometry of $\mathbb{R}$ is present always in one way or other, and the flip of the negative and positive ends usually induces some sort of involution.
