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The unconventional magnetic orderings found in Kitaev spin liquid candidatessuggest frustration of spin interactions, and raise a possibility of nearbyspin liquid phases. In particular, counter-rotating spiral ordering inthree-dimensional (3D) iridates is striking, and understanding the microscopicmechanism of such ordering may provide routes to 3D Kitaev spin liquids. Westudy a minimal 3D model including Kitaev $K$ and a symmetric off-diagonalbond-dependent $\Gamma$ interaction on the hyperhoneycomb lattice using exactdiagonalization. We first show that a 12-site unitary transformation unveils aHeisenberg model with hidden SU(2) symmetry when $K = \Gamma$. The magneticordering in the transformed basis then generically maps to the counter-rotatingnoncoplanar spiral order of the original spin. The moment direction depends onperturbations away from the SU(2) point. When $K$ and $\Gamma$ are negative, apositive Heisenberg interaction favors the (110)-direction reported in theneutron scattering measurement on $\beta$-Li$_2$IrO$_3$. Our findings offer arelevant set of microscopic parameters, which in turn guides a way to approachpossible Kitaev spin liquids.

Counter-rotating spiral order in three dimensional iridates: signature of hidden symmetry in Kitaev-$\Gamma$ model