  We discuss the double copy formulation of Moyal-Weyl type noncommutative
gauge theories from the homotopy algebraic perspective of factorisations of
$L_\infty$-algebras. We define new noncommutative scalar field theories with
rigid colour symmetries taking the role of the zeroth copy, where the deformed
colour algebra plays the role of a kinematic algebra; some of these theories
have a trivial classical limit but exhibit colour-kinematics duality, from
which we construct the double copy theory explicitly. We show that
noncommutative gauge theories exhibit a twisted form of colour-kinematics
duality, which we use to show that their double copies match with the
commutative case. We illustrate this explicitly for Chern-Simons theory, and
also for Yang-Mills theory where we obtain a modified Kawai-Lewellen-Tye
relation whose momentum kernel is linked to a binoncommutative biadjoint scalar
theory. We reinterpret rank one noncommutative gauge theories as double copy
theories, and discuss how our findings tie in with recent discussions of
Moyal-Weyl deformations of self-dual Yang-Mills theory and gravity.
