  The Kitaev material $\alpha$-RuCl$_3$ is among the most prominent candidates
to host a quantum spin-liquid state endowed with fractionalized excitations.
Recent experimental and theoretical investigations have separately revealed the
importance of both the magnetoelastic coupling and the magnetic anisotropy, in
dependence of the applied magnetic field direction. In this combined
theoretical and experimental research, we investigate the anisotropic magnetic
and magnetoelastic properties for magnetic fields applied along the main
crystallographic axes as well as for fields canted out of the honeycomb plane.
We found that the magnetostriction anisotropy is unusually large compared to
the anisotropy of the magnetization, which is related to the strong
magnetoelastic $\widetilde{\Gamma'}$-type coupling in our \textit{ab-initio}
derived model. We observed large, non-symmetric magnetic anisotropy for
magnetic fields canted out of the honeycomb $ab$-plane in opposite directions,
namely towards the $+c^*$ or $-c^*$ axes, respectively. The observed
directional anisotropy is explained by considering the relative orientation of
the magnetic field with respect to the co-aligned RuCl$_6$ octahedra.
Magnetostriction measurements in canted fields support this non-symmetric
magnetic anisotropy, however these experiments are affected by magnetic torque
effects. Comparison of theoretical predictions with experimental findings allow
us to recognize the significant contribution of torque effects in experimental
setups where $\alpha$-RuCl$_3$ is placed in canted magnetic fields.
