We have conducted high-pressure x-ray diffraction and Raman spectroscopic studies on the $\mathrm{CdC}{\mathrm{r}}_{2}\mathrm{S}{\mathrm{e}}_{4}$ spinel at room temperature up to 42 GPa. We have resolved three structural transitions up to 42 GPa, i.e., the starting $Fd\overline{3}m$ phase transforms at $\ensuremath{\sim}11\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ into a tetragonal $I{4}_{1}/amd$ structure, an orthorhombic distortion was observed at $\ensuremath{\sim}15\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$, whereas structural disorder initiates beyond 25 GPa. Our ab initio density functional theory studies successfully reproduced the observed crystalline-to-crystalline structural transitions. In addition, our calculations propose an antiferromagnetic ordering as a potential magnetic ground state for the high-pressure tetragonal and orthorhombic modifications, compared with the starting ferromagnetic phase. Furthermore, the computational results indicate that all phases remain insulating in their stability pressure range, with a direct-to-indirect band gap transition for the $Fd\overline{3}m$ phase taking place at 5 GPa. We attempted also to offer an explanation behind the peculiar first-order character of the $Fd\overline{3}m(\mathrm{cubic})\ensuremath{\rightarrow}I{4}_{1}/amd$ (tetragonal) transition observed for several relevant Cr spinels, i.e., the sizeable volume change at the transition point, which is not expected from space group symmetry considerations. We detected a clear correlation between the cubic-tetragonal transition pressures and the next-nearest-neighbor magnetic exchange interactions for the Cr-bearing sulfide and selenide members, a strong indication that the cubic-tetragonal transitions in these systems are principally governed by magnetic effects.