  We study the hypersymplectic spaces obtained as quotients of flat
hypersymplectic space R^{4d} by the action of a compact Abelian group. These
4n-dimensional quotients carry a multi-Hamilitonian action of an n-torus. The
image of the hypersymplectic moment map for this torus action may be described
by a configuration of solid cones in R^{3n}. We give precise conditions for
smoothness and non-degeneracy of such quotients and show how some properties of
the quotient geometry and topology are constrained by the combinatorics of the
cone configurations. Examples are studied, including non-trivial structures on
R^{4n} and metrics on complements of hypersurfaces in compact manifolds.
