TITLE: Is there a deep reason for the fecundity of involutions?
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: 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.