$Q$-shaped derived categories as derived categories of differential graded bimodules

We prove that, under mild assumptions, the $Q$-shaped derived categories introduced by Holm and Jørgensen are equivalent to derived categories of differential graded bimodules over differential graded categories. This yields new derived invariance results for $Q$-shaped derived categories that allow us to extend known descriptions of such categories as derived categories of differential graded bimodules over (possibly graded) algebras.