  Applying a classical theorem of Smith, we show that the poset property of
being Gorenstein$^*$ over $\mathbb{Z}_2$ is inherited by the subposet of fixed
points under an involutive poset automorphism. As an application, we prove that
every interval in the Bruhat order on (twisted) involutions in an arbitrary
Coxeter group has this property, and we find the rank function. This implies
results conjectured by F. Incitti. We also show that the Bruhat order on the
fixed points of an involutive automorphism induced by a Coxeter graph
automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as
a Coxeter group in its own right.
