  With large spin-orbit coupling, the $t_{2g}^5$ electron configuration in
$d$-metal oxides is prone to highly anisotropic exchange interactions and
exotic magnetic properties. In $5d^5$ iridates, given the existing variety of
crystal structures, the magnetic anisotropy can be tuned from antisymmetric to
symmetric Kitaev-type, with interaction strengths that outsize the isotropic
terms. By many-body electronic-structure calculations we here address the
nature of the magnetic exchange and the intriguing spin-glass behavior of
Li$_2$RhO$_3$, a $4d^5$ honeycomb oxide. For pristine crystals without Rh-Li
site inversion, we predict a dimerized ground state as in the isostructural
$5d^5$ iridate Li$_2$IrO$_3$, with triplet spin dimers effectively placed on a
frustrated triangular lattice. With Rh-Li anti-site disorder, we explain the
observed spin-glass phase as a superposition of different, nearly degenerate
symmetry-broken configurations.
