  The orientation of N\'eel-type skyrmions in the lacunar spinels GaV$_4$S$_8$
and GaV$_4$Se$_8$ is tied to the polar axes of their underlying crystal
structure through the Dzyaloshinskii-Moriya interaction. In these crystals, the
skyrmion lattice phase exists for externally applied magnetic fields parallel
to these axes and withstands oblique magnetic fields up to some critical angle.
Here, we map out the stability of the skyrmion lattice phase in both crystals
as a function of field angle and magnitude using dynamic cantilever
magnetometry. The measured phase diagrams reproduce the major features
predicted by a recent theoretical model, including a reentrant cycloidal phase
in GaV$_4$Se$_8$. Nonetheless, we observe a greater robustness of the skyrmion
phase to oblique fields, suggesting possible refinements to the model. Besides
identifying transitions between the cycloidal, skyrmion lattice, and
ferromagnetic states in the bulk, we measure additional anomalies in
GaV$_4$Se$_8$ and assign them to magnetic states confined to polar structural
domain walls.
