How cells break symmetry and organize activity at their edges to move directionally is a fundamental question in cell biology. Physical models of cell motility commonly incorporate gradients of regulatory proteins and/or feedback from the motion itself to describe the polarization of this edge activity. These approaches, however, fail to explain cell behaviour before the onset of polarization. We use polarizing and moving fish epidermal cells as a model system to bridge the gap between cell behaviours before and after polarization. Our analysis suggests a novel and simple principle of self-organizing cell activity, in which local cell-edge dynamics depends on the distance from the cell centre, but not on the orientation with respect to the front-back axis. We validate this principle with a stochastic model that faithfully reproduces a range of cell-migration behaviours. Our findings indicate that spontaneous polarization, persistent motion and cell shape are emergent properties of the local cell-edge dynamics controlled by the distance from the cell centre.