  Bipartitional relations were introduced by Foata and Zeilberger in their
characterization of relations which give rise to equidistribution of the
associated inversion statistic and major index. We consider the natural partial
order on bipartitional relations given by inclusion. We show that, with respect
to this partial order, the bipartitional relations on a set of size $n$ form a
graded lattice of rank $3n-2$. Moreover, we prove that the order complex of
this lattice is homotopy equivalent to a sphere of dimension $n-2$. Each proper
interval in this lattice has either a contractible order complex, or it is
isomorphic to the direct product of Boolean lattices and smaller lattices of
bipartitional relations.As a consequence, we obtain that the M\"obius function
of every interval is 0, 1, or -1. The main tool in the proofs is discrete Morse
theory as developed by Forman, and an application of this theory to order
complexes of graded posets, designed by Babson and Hersh, in the extended form
of Hersh and Welker.
