  It is well known that string theory has a T-duality symmetry relating circle
compactifications of large and small radius. This symmetry plays a foundational
role in string theory. We note here that while T-duality is order two acting on
the moduli space of compactifications, it is order four in its action on the
conformal field theory state space. More generally, involutions in the Weyl
group $W(G)$ which act at points of enhanced $G$ symmetry have canonical lifts
to order four elements of $G$, a phenomenon first investigated by J. Tits in
the mathematical literature on Lie groups and generalized here to conformal
field theory. This simple fact has a number of interesting consequences. One
consequence is a reevaluation of a mod two condition appearing in asymmetric
orbifold constructions. We also briefly discuss the implications for the idea
that T-duality and its generalizations should be thought of as discrete gauge
symmetries in spacetime.
