The ability to coordinate movements involving multiple body parts is fundamental to behavior. How the brain coordinates several body parts with different ranges of motion and reference frames is unknown. Here we investigated this problem by studying the neural circuits and computations underlying coordinated head and eye movements in mice. It is commonly thought that neural control of coordinated movements entails a hierarchical series of computations that progress from specifying higher- to lower-level movement parameters. Here we reveal a different computational logic implemented in the mouse gaze system. We found that single neurons in the mouse superior colliculus (SC) specify a mixture of eye movement endpoints and head movement displacements. They do so by innervating two separate hindbrain populations that transform identical excitatory input into eye movements with fixed endpoints and head movements with fixed displacements. Neural recordings showed that head displacement information is present in SC, whereas saccade information does not emerge until the hindbrain. Thus, displacements for the head and eyes are computed at different anatomical stages and the desired overall movement is nowhere represented. These results reveal a non-hierarchical computational logic for coordinated movements.