We have performed combined experimental and theoretical high-pressure investigations on magnetoelectric HgCr₂S₄ spinel. Overall, HgCr₂S₄ exhibits three reversible structural transitions under pressure: the starting Fd-3m phase adopts a tetragonal I4₁/amd structure at 20 GPa, an orthorhombic distortion occurs above 26 GPa, whereas a third structural transition takes place beyond 37 GPa. During the Fd-3m to I4₁/amd structural transition, HgCr₂S₄ experiences a volume change of 4% which is unexpected from space group symmetry considerations alone. Furthermore, the Fd-3m to I4₁/amd transformation appears to be concomitant with an insulator-to-metal transition whereas compression of HgCr₂S₄ leads to a gradual suppression of its ferromagnetism.