Modern multi-agent systems frequently use high-level planners to extract basic paths for agents, and then rely on local collision avoidance to ensure that the agents reach their destinations without colliding with one another or dynamic obstacles. One state-of-the-art local collision avoidance technique is Optimal Reciprocal Collision Avoidance (ORCA). Despite being fast and efficient for circular-shaped agents, ORCA may deadlock when polygonal shapes are used. To address this shortcoming, we introduce Reciprocally-Rotating Velocity Obstacles (RRVO). RRVO generalizes ORCA by introducing a notion of rotation for polygonally-shaped agents. This generalization permits more realistic motion than ORCA and does not suffer from as much deadlock. In this paper, we present the theory of RRVO and show empirically that it does not suffer from the deadlock issue ORCA has, permits agents to reach goals faster, and has a comparable collision rate at the cost of performance overhead quadratic in the (typically small) user-defined parameter δ.