  Landau level spectroscopy has been employed to probe the electronic structure
of the valence band in a series of p-type HgTe/HgCdTe quantum wells with both
normal and inverted ordering of bands. We find that the standard
axial-symmetric 4-band Kane model, which is nowadays widely applied in physics
of HgTe-based topological materials, does not fully account for the complex
magneto-optical response observed in our experiments - notably, for the
unexpected avoided crossings of excitations and for the appearance of
transitions that are electric-dipole forbidden within this model. Nevertheless,
reasonable agreement with experiments is achieved when the standard model is
expanded to include effects of bulk and interface inversion asymmetries. These
remove the axial symmetry, and among other, profoundly modify the shape of
valence bands.
