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Propulsion via cyclic moves presents a challenge to micro-objects in overdamped environments, where time reversibility may prevent directed motion. Most reported cyclic movers exploit anisotropic hydrodynamic drag with the surrounding medium to push or pull themselves forward. Here, we present a propulsion strategy that exploits internal interactions in multicomponent objects. We study self-assembled clusters of magnetic particles that exhibit an off-center dipole moment. By theoretical modeling and in experiments with magnetic Janus particles, we demonstrate that under an oscillating field the interaction between such anisotropic particles in the cluster breaks time reversibility. This enables propulsion under an oscillating actuation field without changing the external shape of the object, which is impossible within the strategy of hydrodynamic drag. Using the same experimental particle system, we realize various modes of motion ranging from stirrers to steerable movers with helical or directed path under the same environmental conditions. They present a powerful counterpart to propellers that move via hydrodynamic drag.