  We prove that the trace of the $n$th power of the antipode of a Hopf algebra
with the Chevalley property is a gauge invariant, for each integer $n$. As a
consequence, the order of the antipode, and its square, are invariant under
Drinfeld twists. The invariance of the order of the antipode is closely related
to a question of Shimizu on the pivotal covers of finite tensor categories,
which we affirmatively answer for representation categories of Hopf algebras
with the Chevalley property.
