  We prove that the subposet induced by the fixed elements of any automorphism
of a pircon is also a pircon. By a result of Abdallah, Hansson, and Hultman,
the order complex of any open interval in a pircon is a PL ball or a PL sphere.
We apply our main results to symmetric groups of the form $S_{2n}$. A
consequence is that the fixed point free signed involutions form a pircon under
the dual of the Bruhat order on the hyperoctahedral group. Finally, we prove
that this poset is, in fact, EL-shellable, which is a type $B$ analogue of a
result of Can, Cherniavsky, and Twelbeck.
