@NickGill In reply to your first comment, what's going on is that Dirac fermions in two dimensions can be gapped out by interactions preserving $\mathbb{Z}_2$ axial fermion parity only in multiples of 4, a story which began here. The question is a translation of this fact in which the matrix $R$ plays the role of the interactions.

Thanks, I was hoping this question might get your attention. Along similar lines, there's also this paper, which counts the number of possible lattices $\Lambda$ of a given volume, for $N \leq 4$. But as the motivations of the paper are different, it doesn't appear to address my question: when is $\Lambda$ even?