  The unconventional magnetic orderings found in Kitaev spin liquid candidates
suggest frustration of spin interactions, and raise a possibility of nearby
spin liquid phases. In particular, counter-rotating spiral ordering in
three-dimensional (3D) iridates is striking, and understanding the microscopic
mechanism of such ordering may provide routes to 3D Kitaev spin liquids. We
study a minimal 3D model including Kitaev $K$ and a symmetric off-diagonal
bond-dependent $\Gamma$ interaction on the hyperhoneycomb lattice using exact
diagonalization. We first show that a 12-site unitary transformation unveils a
Heisenberg model with hidden SU(2) symmetry when $K = \Gamma$. The magnetic
ordering in the transformed basis then generically maps to the counter-rotating
noncoplanar spiral order of the original spin. The moment direction depends on
perturbations away from the SU(2) point. When $K$ and $\Gamma$ are negative, a
positive Heisenberg interaction favors the (110)-direction reported in the
neutron scattering measurement on $\beta$-Li$_2$IrO$_3$. Our findings offer a
relevant set of microscopic parameters, which in turn guides a way to approach
possible Kitaev spin liquids.
