Terrestrial particle accelerators collide charged particles, then watch the trajectory of outgoing debris - but they cannot manipulate dark matter. Fortunately, dark matter is the main component of galaxy clusters, which are continuously pulled together by gravity. We show that galaxy cluster mergers can be exploited as enormous, natural dark matter colliders. We analyse hydrodynamical simulations of a universe containing self-interacting dark matter (SIDM) in which all particles interact via gravity, and dark matter particles can also scatter off each other via a massive mediator. During cluster collisions, SIDM spreads out and lags behind cluster member galaxies. Individual systems can have quirky dynamics that makes them difficult to interpret. Statistically, however, we find that the mean or median of dark matter's spatial offset in many collisions can be robustly modelled, and is independent of our viewing angle and halo mass even in collisions between unequal-mass systems. If the SIDM cross-section were sigma/m = 0.1 cm(2) g(-1) = 0.18 barn GeV-1, the 'bulleticity' lag would be similar to 5 per cent that of gas due to ram pressure, and could be detected at 95 per cent confidence level in weak lensing observations of similar to 100 well-chosen clusters.