Distribution of rooks on a chess-board representing a Latin square partitioned by a subsystem

A d-dimensional generalization of a Latin square of order n is a chess-board of size n x n x ... x n (d times) carrying n^(d-1) non-attacking rooks, equivalently a high-dimensional permutation in the sense of Linial and Luria. For a subsystem T induced by a family (E_1, ..., E_d) of subsets of {1,2,...,n}, let df(T) = V(T)/n - c, where V(T) is the number of cells of T and c the number of rooks in it.
Replacing k of the sets E_i by their complements yields a subsystem T_k, and the 2^d subsystems so obtained partition the chess-board. We prove that df(T_k) = (-1)^k df(T_0), and give a multilinear proof showing that no assumption on the sets E_i is needed and that the result holds verbatim for d-fold stochastic matrices. Consequently the partition has a single degree of freedom: the rook counts of all its 2^d members follow from their volumes and the single number df(T_0). For d = k = 3 this specializes to the identity of Cruse relating a brick and its remote mate, a necessary condition for a partial Latin square to be completable. We also prove that for d = 3 the chess-board represents precisely one main class.