When a principal with private information designs a mechanism to coordinate his subordinates, he faces a dilemma: to conceal his information, his selection of mechanism must not depend on his information; but his information may influence which mechanism he prefers. To resolve this dilemma, this paper develops a theory of inscrutable mechanism selection. The principal's neutral optima are defined as the smallest possible set of unblocked mechanisms. They are shown to exist and are characterized using parametric linear programs. Any safe and undominated mechanism is a neutral optimum. Any neutral optimum is an expectional equilibrium and a core mechanism.