  A non-empty subset A of X = X_1 x...x X_d is a (proper) box if A = A_1 x...x
A_d and A_i is a (proper) subset of X_i for each i. Suppose that for each pair
of boxes A, B and each i, one can only know which of the three states takes
place: A_i = B_i, A_i = X_i\setminus B_i, A_i is different from each of the
sets B_i, X_i\setminus B_i. Let F and G be two systems of disjoint boxes. Can
one decide whether their unions are equal? In general, the answer is no, but as
is shown in the paper, it is yes if both systems consist of pairwise
dichotomous boxes. Several criteria that enable to compare such systems are
collected. The paper includes also rigidity results, which say what assumptions
have to be imposed on F to ensure that \bigcup F = \bigcup G implies F = G. As
an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias
and Shor is verified.
