  We show that, if $\Gamma$ is a point group of $\mathbb{R}^{k+1}$ of order two
for some $k\geq 2$ and $\mathcal S$ is a $k$-pseudomanifold which has a free
automorphism of order two, then either $\mathcal S$ has a $\Gamma$-symmetric
infinitesimally rigid realisation in $\mathbb{R}^{k+1}$ or $k=2$ and $\Gamma$
is a half-turn rotation group.This verifies a conjecture made by Klee, Nevo,
Novik and Zhang for the case when $\Gamma$ is a point-inversion group. Our
result implies that Stanley's lower bound theorem for centrally symmetric
polytopes extends to pseudomanifolds with a free simplicial involution, thus
verifying (the inequality part) of another conjecture of Klee, Nevo, Novik and
Zheng. Both results actually apply to a much larger class of simplicial
complexes, namely the circuits of the simplicial matroid. The proof of our
rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric
simplicial complexes.
