Horizontal and vertical movements of the human eye bring new objects to the center of the visual field, but torsional movements rotate the visual world about its center. Ocular torsion stays near zero during head-fixed gaze shifts, and eye movements to Visual targets are thought to be driven by purely horizontal and vertical commands. Here, analysis of eye-head gaze shifts revealed that gaze commands were three-dimensional, with a separate neural control system for torsion. Active torsion optimized gaze control as no two-dimensional system could have, stabilizing the retinal image as quickly as possible when it would otherwise have spun around the fixation point.