  Following an idea of A. Berenstein, we define a commutor for the category of
crystals of a finite dimensional complex reductive Lie algebra. We show that
this endows the category of crystals with the structure of a coboundary
category. Similar to the case of braided categories, there is a group naturally
acting on multiple tensor products in coboundary categories. We call this group
the cactus group and identify it as the fundamental group of the moduli space
of marked real genus zero stable curves.
