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We employed resonant X-ray magnetic diffraction to directly probe the evolution with pressure of all-in-all-out (AIAO) antiferromagnetic order in the cubic pyrochlore Cd₂Os₂O₇. The AIAO order is suppressed through a continuous quantum phase transition with neither an abrupt change in the electronic configuration nor a discontinuity in the lattice constant at a critical pressure, P_c=36.5GPa. Concomitant with the recovery of time-reversal symmetry in the magnetically disordered state, the crystal lattice experiences a spontaneous inversion symmetry breaking with the symmetry group continuously changing from Fd-3m to F-43m. An insulator-metal transition appears to accompany the magnetic and structural transitions. As spin fluctuations, lattice breathing modes, and quasiparticle excitations interact in the quantum critical region, we argue that they present the necessary components for strongly-coupled quantum criticality in this three-dimensional magnet.