Cytoskeletal motors drive the transport of organelles and molecular cargoes within cells and have potential applications in molecular detection and diagnostic devices. Engineering molecular motors with controllable properties will allow selective perturbation of mechanical processes in living cells and provide optimized device components for tasks such as molecular sorting and directed assembly. Biological motors have previously been modified by introducing activation/deactivation switches that respond to metal ions and other signals. Here, we show that myosin motors can be engineered to reversibly change their direction of motion in response to a calcium signal. Building on previous protein engineering studies and guided by a structural model for the redirected power stroke of myosin VI, we have constructed bidirectional myosins through the rigid recombination of structural modules. The performance of the motors was confirmed using gliding filament assays and single fluorophore tracking. Our strategy, in which external signals trigger changes in the geometry and mechanics of myosin lever arms, should make it possible to achieve spatiotemporal control over a range of motor properties including processivity, stride size and branchpoint turning.