We identify a new phenomenon in neural network optimization which arises from the interaction of depth and a particular heavy-tailed structure in natural data. Our result offers intuitive explanations for several previously reported observations about network training dynamics, including a conceptually new cause for progressive sharpening and the edge of stability. We further draw connections to related phenomena including grokking and simplicity bias.Experimentally, we demonstrate the significant influence of paired groups of outliers in the training data with strong Opposing Signals: consistent, large magnitude features which dominate the network output and provide gradients which point in opposite directions. Due to these outliers, early optimization enters a narrow valley which carefully balances the opposing groups; subsequent sharpening causes their loss to rise rapidly, oscillating between high on one group and then the other, until the overall loss spikes. We carefully study these groups' effect on the network's optimization and behavior, and we complement this with a theoretical analysis of a two-layer linear network under a simplified model.Our finding enables new qualitative predictions of training behavior which we confirm experimentally. It also provides a new lens through which to study and improve modern training practices for stochastic optimization, which we highlight via a case study of Adam versus SGD.