To explore the visible world, human beings and other primates often rely on gaze shifts. These are coordinated movements of the eyes and head characterized by stereotypical metrics and kinematics. It is possible to determine the rules that the effectors must obey to execute them rapidly and accurately and the neural commands needed to implement these rules with the help of optimal control theory. In this study, we demonstrate that head-fixed saccades and head-free gaze shifts obey a simple physical principle, "the minimum effort rule." By direct comparison with existing models of the neural control of gaze shifts, we conclude that the neural circuitry that implements the minimum effort rule is one that uses inhibitory cross talk between independent eye and head controllers.