  We discuss magnetically ordered states, arising in Heisenberg-Kitaev and
related spin models, on three-dimensional (3D) harmonic honeycomb lattices. For
large classes of ordered states, we show that they can be mapped onto
two-dimensional (2D) counterparts on the honeycomb lattice, with the classical
energetics being identical in the 2D and 3D cases. As an example, we determine
the phase diagram of the classical nearest-neighbor Heisenberg-Kitaev model on
the hyperhoneycomb lattice in a magnetic field: This displays rich and complex
behavior akin to its 2D counterpart, with most phases and phase boundaries
coinciding exactly. To make contact with the physics of the hyperhoneycomb
iridate $\beta$-Li$_2$IrO$_3$, we also include a symmetric off-diagonal
$\Gamma$ interaction, discuss its 3D-2D mapping, and determine the relevant
phase diagrams. In particular, we demonstrate explicitly the adiabatic
equivalence of the spiral magnetic orders in $\alpha$- and
$\beta$-Li$_2$IrO$_3$. Our results pave the way to a systematic common
understanding of 2D and 3D Kitaev materials.
