  Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in
six dimensions, and we use this correspondence to interpret symplectic
fibrations between these orbits, and to analyse moment polytopes associated to
the standard Hamiltonian torus action on the coadjoint orbits. The theory is
then applied to describe so-called intrinsic torsion varieties of Riemannian
structures on the Iwasawa manifold.
