Recent efforts to create challenge benchmarks that test the abilities of natural language understanding models have largely depended on human annotations. In this work, we introduce the "Break, Perturb, Build" (BPB) framework for automatic reasoning-oriented perturbation of question-answer pairs. BPB represents a question by decomposing it into the reasoning steps that are required to answer it, symbolically perturbs the decomposition, and then generates new question-answer pairs. We demonstrate the effectiveness of BPB by creating evaluation sets for three reading comprehension (RC) benchmarks, generating thousands of high-quality examples without human intervention. We evaluate a range of RC models on our evaluation sets, which reveals large performance gaps on generated examples compared to the original data. Moreover, symbolic perturbations enable fine-grained analysis of the strengths and limitations of models. Last, augmenting the training data with examples generated by BPB helps close the performance gaps, without any drop on the original data distribution.