  Motivated by the recent result that left-orderability of a group $G$ is
intimately connected to circular orderability of direct products $G \times
\mathbb{Z}/n\mathbb{Z}$, we provide necessary and sufficient cohomological
conditions that such a direct product be circularly orderable. As a consequence
of the main theorem, we arrive at a new characterization for the fundamental
group of a rational homology 3-sphere to be left-orderable. Our results imply
that for mapping class groups of once-punctured surfaces, and other groups
whose actions on $S^1$ are cohomologically rigid, the products $G \times
\mathbb{Z}/n\mathbb{Z}$ are seldom circularly orderable. We also address
circular orderability of direct products in general, dealing with the cases of
factor groups admitting a bi-invariant circular ordering, and iterated direct
products whose factor groups are amenable.
