  We examine the topologically twisted index of $\mathcal N=4$ super-Yang-Mills
with gauge group $SU(N)$ on $T^2\times S^2$, and demonstrate that it receives
contributions from multiple sectors corresponding to the freely acting
orbifolds $T^2/\mathbb Z_m\times\mathbb Z_n$ where $N=mn$. After summing over
these sectors, the index can be expressed as the elliptic genus of a
two-dimensional $\mathcal N=(0,2)$ theory resulting from Kaluza-Klein reduction
on $S^2$. This provides an alternate path to the 'high-temperature' limit of
the index, and confirms the connection to the right-moving central charge of
the $\mathcal N=(0,2)$ theory.
