A pair of bubbles starting from rest and rising side-by-side in a liquid have been shown earlier to display spherical and ellipsoidal shapes. In contrast to earlier computational studies on the two-dimensional dynamics of a pair of bubbles, we study the fully three-dimensional motion of the bubbles in the inertial regime. We reveal the destabilizing nature of the interaction between the wakes of the bubbles, which causes them to rise in an oscillatory path. Such three-dimensionality sets in earlier in time than for a single bubble and also at a lower inertia. The interaction leads to a mirror symmetry in the trajectories of the two bubbles, which persists for some time even in the high-inertia regime where each path is chaotic. The effect of the inertia and initial separation on the mirror symmetry of the path, the vortex shedding pattern, and the attraction and repulsion between the bubbles are examined. The bubble rise has been interestingly observed to be symmetrical about the plane perpendicular to the separation vector for all separation distances considered in the present study. © 2017 American Physical Society.