We consider a general mechanism-design environment in which the planner faces incentive constraints such as the ones resulting from agents' private information or ability to take hidden actions. We study the properties of optimal mechanisms when some decisions are incentive-separable: A set of decisions is incentive-separable if, starting at some initial allocation, perturbing these decisions along agents' indifference curves preserves incentive constraints. We show that, under regularity conditions, the optimal mechanism allows agents to make unrestricted choices over incentive-separable decisions, given some prices and budgets. Using this result, we extend and unify the Atkinson-Stiglitz theorem on the undesirability of differentiated commodity taxes and the Diamond-Mirrlees production efficiency result. We also demonstrate how the analysis of incentive separability can provide a novel justification for in-kind redistribution programs similar to food stamps.