We examine the distribution and characteristics of 70 reverse-faulting earthquakes at slowly-spreading ridges in the Atlantic and Indian Oceans. All were large enough (Mw≥4.7) for their focal mechanisms to be determined and listed in the gCMT catalogue (www.globalcmt.org/). Of these, 21 were sufficiently large (Mw≥5.3) for detailed body-waveform analysis to determine accurate mechanisms, centroid depths and water depths at their epicentres. All had shallow centroids 1–3 km below the sea bed, and occurred beneath gravity and bathymetric highs on ridge flanks, not in the axial valley, consistent with the locations of reverse-faulting microearthquakes discovered in earlier OBS experiments. We describe a simple quantitative model that explains their occurrence by the bending of an elastic layer about 3 km thick as it is uplifted and exhumed into the dome of a core complex by detachment faulting. The widespread distribution of such events, and their global seismic moment-release rate, suggests that asymmetric spreading in core complexes is the dominant mode of extension at such slowly-spreading ridges. Such an inherently asymmetric process is capable of producing a steady-state axial rift valley, and the overall symmetry of plate creation, but only if its polarity reverses in space and time along the axis of the ridge, for which there is some evidence in the gravity from altimetry.