Combinatorial spin structures, update

In 2d spin structures correspond to Kasteleyn orientations on a graph (i.e. orientations of edges so that every face has an odd number of clockwise oriented edges). Hat tip to jedal This fact is important in the theory of dimer models. Dimer model is related to the 2d Ising model, and the fact that its partition function is expressed as a sum over equivalence classes of Kasteleyn orientations is related to the fact that the critical Ising model is equivalent to a free fermion model, and the latter involves summing over spin structures.

In general (for arbitrary dimensions) a construction is proposed in http://arxiv.org/pdf/1306.4841.pdf   but it is horribly complicated, I do not even see how it is related to Kasteleyn orientations in the 2d case.

Upd. Asked the question on mathoverflow: http://mathoverflow.net/questions/143067/combinatorial-spin-structures