  Solutions to the strong CP problem typically introduce new scales associated
with the spontaneous breaking of symmetries. Absent any anthropic argument for
small $\bar\theta$, these scales require stabilization against ultraviolet
corrections. Supersymmetry offers a tempting stabilization mechanism, since it
can solve the "big" electroweak hierarchy problem at the same time. One family
of solutions to strong CP, including generalized parity models, heavy axion
models, and heavy $\eta^\prime$ models, introduces $\mathbb{Z}_2$ copies of
(part of) the Standard Model and an associated scale of
$\mathbb{Z}_2$-breaking. We review why, without additional structure such as
supersymmetry, the $\mathbb{Z}_2$-breaking scale is unacceptably tuned. We then
study "SUZ$_2$" models, supersymmetric theories with $\mathbb{Z}_2$ copies of
the MSSM. We find that the addition of SUSY typically destroys the
$\mathbb{Z}_2$ protection of $\bar\theta=0$, even at tree level, once SUSY and
$\mathbb{Z}_2$ are broken. In theories like supersymmetric completions of the
twin Higgs, where $\mathbb{Z}_2$ addresses the little hierarchy problem but not
strong CP, two axions can be used to relax $\bar\theta$.
