Making controlled perturbations is essential for various tasks (e.g., data augmentation), but building task-specific generators can be expensive. We introduce Tailor, a taskagnostic generation system that perturbs text in a semantically-controlled way. With unlikelihood training, we design Tailor's generator to follow a series of control codes derived from semantic roles. Through modifications of these control codes, Tailor can produce fine-grained perturbations. We implement a set of operations on control codes that can be composed into complex perturbation strategies, and demonstrate their effectiveness in three distinct applications: First, Tailor facilitates the construction of high-quality contrast sets that are lexically diverse, and less biased than original task test data. Second, paired with automated labeling heuristics, Tailor helps improve model generalization through data augmentation: We obtain an average gain of 1.73 on an NLI challenge set by perturbing just ≈5% of training data. Third, without any finetuning overhead, Tailor's perturbations effectively improve compositionality in fine-grained style transfer, outperforming fine-tuned baselines on 6 transfers.