  In a recent paper, we have shown that warped time-frequency representations
provide a rich framework for the construction and study of smoothness spaces
matched to very general phase space geometries obtained by diffeomorphic
deformations of $\mathbb{R}^d$. Here, we study these spaces, obtained through
the application of general coorbit theory, using the framework of decomposition
spaces. This allows us to derive embedding relations between coorbit spaces
associated to different warping functions, and relate them to established,
important smothness spaces. In particular, we show that we obtain
$\alpha$-modulation spaces and spaces of dominating mixed smoothness as special
cases and, in contrast, that this is only possible for Besov spaces if $d=1$.
