Band inversion is a unique physical phenomenon in topological insulators that affects the bulk electronic structure. In particular, band inversion may induce warping effects on the bulk band structure leading to high valley degeneracy. We find that there are two rules under which band inversion leads to warped bands in the bulk: (i) the bands must be sufficiently inverted, and (ii) secondary bands must be close in energy to, and have opposite parity from, the band edges. We clarify that band inversion is a manifestation of both spin-orbit interactions and atomic orbital interactions, allowing multiple routes for rationally tuning the band inversion in a material. We tabulate the possible band inversion-induced valley degeneracies in all 11 centrosymmetric point groups, indicating that a surge in the valley degeneracy from 2-fold up to 48-fold at a high-symmetry k-point is possible due to band inversion. The present study therefore offers a unique avenue for engineering the electronic structures of materials, either by inverting bands that are normally non-inverted or by modifying the band inversion effects in topological insulators with bands that are already inverted.