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 in optimization including grokking and simplicity bias.Experimentally, we demonstrate the significant influence of paired groups of outliers in the training data with strong \emph{opposing signals}: consistent, large magnitude features which dominate the network output and occur in both groups with similar frequency.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 complement these experiments with a theoretical analysis of a two-layer linear network on a simple model of opposing signals. 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. For instance, we identify two small modifications to Momentum SGD which result in performance that matches adaptive methods in settings where it has traditionally faltered---including on attention models.