We introduce structured projection of intermediate gradients (SPIGOT), a new method for backpropagating through neural networks that include hard-decision structured predictions (e.g., parsing) in intermediate layers. SPIGOT requires no marginal inference, unlike structured attention networks and reinforcement learning-inspired solutions. Like so-called straight-through estimators, SPIGOT defines gradient-like quantities associated with intermediate nondifferentiable operations, allowing backpropagation before and after them; SPIGOT's proxy aims to ensure that, after a parameter update, the intermediate structure will remain well-formed. We experiment on two structured NLP pipelines: syntactic-then-semantic dependency parsing, and semantic parsing followed by sentiment classification. We show that training with SPIGOT leads to a larger improvement on the downstream task than a modularly-trained pipeline, the straight-through estimator, and structured attention, reaching a new state of the art on semantic dependency parsing.