  A permutation is centrosymmetric if it is fixed by a half-turn rotation of
its diagram. Initially motivated by a question by Alexander Woo, we investigate
the question of whether the growth rate of a permutation class equals the
growth rate of its even-size centrosymmetric elements. We present various
examples where the latter growth rate is strictly less, but we conjecture that
the reverse inequality cannot occur. We conjecture that equality holds if the
class is sum closed, and we prove this conjecture in the special case where the
growth rate is at most $\xi \approx 2.30522$, using results from Pantone and
Vatter on growth rates less than $\xi$. We prove one direction of inequality
for sum closed classes and for some geometric grid classes. We end with
preliminary findings on new kinds of growth-rate thresholds that are a little
bit larger than $\xi$.
