We discuss how Moyal deformations of gauge theories, which arise naturally from open string theory, fit into the paradigm of colour-kinematics duality and the double copy of gauge theory to gravity. Along the way we encounter novel noncommutative scalar field theories with rigid colour symmetry that have no interacting commutative counterparts. These scalar theories offer new perspectives on old ideas that rank one noncommutative gauge theories are gravitational theories. This is rendered explicit in four dimensions where they and their double copy images yield deformations of integrable theories describing the self-dual sectors of Yang-Mills theory and gravity.