  The Kitaev-Heisenberg (KH) model has been proposed to capture magnetic
interactions in iridate Mott insulators on the honeycomb lattice. We show that
analogous interactions arise in many other geometries built from edge-sharing
IrO_6 octahedra, including the pyrochlore and hyperkagome lattices relevant to
Ir2O4 and Na4Ir3O8 respectively. The Kitaev spin liquid exact solution does not
generalize to these lattices. However, a different exactly soluble point of the
honeycomb lattice KH model, obtained by a four-sublattice transformation to a
ferromagnet, generalizes to all these lattices. A Klein four-group =Z2xZ2
structure is associated with this mapping (hence Klein duality). A finite
lattice admits the duality if a simple geometrical condition is met. This
duality predicts fluctuation free ordered states on these different 2D and 3D
lattices, which are analogs of the honeycomb lattice KH stripy order. This
result is used in conjunction with a semiclassical Luttinger-Tisza
approximation to obtain phase diagrams for KH models on the different lattices.
We also discuss a Majorana fermion based mean field theory at the Kitaev point,
which is exact on the honeycomb lattice, for the KH models on the different
lattices. We attribute the rich behavior of these models to the interplay of
geometric frustration and frustration induced by spin-orbit coupling.
