  We obtain the most general forms of rank-2 and rank-3 tensors allowed by the
crystal symmetries of the honeycomb lattice of edge-sharing octahedra for
crystals belonging to different crystallographic point groups, including the
monoclinic point group $2/m$ and the trigonal (or rhombohedral) point group
$\bar{3}$. Our results are relevant for two-dimensional materials, such as
$\alpha$-RuCl$_3$, CrI$_3$, and the honeycomb iridates. We focus on the
magnetic-field-dependent thermal conductivity tensor $\kappa_{ij}(\mathbf{H})$,
which describes a system's longitudinal and thermal Hall responses, for the
cases when the magnetic field is applied along high-symmetry directions,
perpendicular to the plane and in the plane. We highlight some unexpected
results, such as the equality of fully-longitudinal components to
partially-transverse components in rank-3 tensors for systems with three-fold
rotational symmetry, and make testable predictions for the thermal conductivity
tensor.
