  We consider the analyzing power on a spin-0 nuclear target. This observable
is related to the imaginary part of the two-photon-exchange (box) diagram. We
consider the contributions of elastic and inelastic intermediate states. The
former requires knowledge of the elastic nuclear form factor, while the latter
uses the optical theorem as input. The elastic contribution scales as the
nuclear charge $Z$, while the inelastic contribution as the ratio of the atomic
number and nuclear charge, $A/Z$. We provide estimates for $^4$He and
$^{208}$Pb, in the kinematics of existing or upcoming experiments. In both
cases, we predict negative values of a few parts per million, and the dominant
contribution is due to inelastic contributions. The analyzing power can
contribute a substantial systematic error in parity-violating experiments.
