Supercontinents episodically assemble and break up, in association with the closure and opening of ocean basins. During these cycles, continental margins are repeatedly weakened and deformed during subduction, orogeny and rifting, whereas continental cores tend to remain intact. It has therefore been suggested that deformation during supercontinent cycles is controlled by the pre-existing structure of the lithosphere, for example by rheological heterogeneities and mechanical anisotropy that were acquired during past tectonic events. However, observational constraints for this idea have been lacking. Here we present global, high-resolution maps of the lithosphere’s effective elastic thickness over the continents—a proxy for the rigidity or long-term strength of the lithosphere—calculated from a comparison of the spectral coherence between topography and gravity anomalies and the flexural response of an equivalent elastic plate to loading. We find that effective elastic thickness is high in Archean cratons, but low in the surrounding Phanerozoic belts. We also estimate the anisotropy in effective elastic thickness, indicative of a directional dependence of lithospheric rigidity, and show that directions of mechanical weakness align with large gradients in effective elastic thickness and with tectonic boundaries. Our findings support the notion that lithospheric rigidity is controlled by pre-existing structure, and that during the supercontinent cycle, strain is concentrated at pre-existing zones of weakness.