The abyssal ocean is broadly characterized by northward flow of the densest waters and southward flow of less-dense waters above them. Understanding what controls the strength and structure of these interhemispheric flows-referred to as the abyssal overturning circulation-is key to quantifying the ocean's ability to store carbon and heat on timescales exceeding a century. Here we show that, north of 32 degrees S, the depth distribution of the seafloor compels dense southernorigin waters to flow northward below a depth of about 4 kilometres and to return southward predominantly at depths greater than 2.5 kilometres. Unless ventilated from the north, the overlying mid-depths (1 to 2.5 kilometres deep) host comparatively weak mean meridional flow. Backed by analysis of historical radiocarbon measurements, the findings imply that the geometry of the Pacific, Indian and Atlantic basins places a major external constraint on the overturning structure.