  This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue
spaces with respect to the quasi-regular representation of the semi-direct
product of Euclidean space and a suitable matrix dilation group. The class of
dilation groups that we allow, the so-called integrably admissible dilation
groups, contains the matrix groups yielding an irreducible, square-integrable
quasi-regular representation as a proper subclass. The obtained scale of
coorbit spaces extends therefore the well-studied wavelet coorbit spaces
associated to discrete series representations. We show that for any integrably
admissible dilation group there exists a convienent space of smooth, admissible
analyzing vectors that can be used to define a consistent coorbit space
possessing all the essential properties that are known to hold in the setting
of discrete series representations. In particular, the obtained coorbit spaces
can be realized as Besov-type decomposition spaces by means of a
Littlewood-Paley type characterization. The classes of anisotropic Besov spaces
associated to expansive matrices are shown to coincide precisely with the
coorbit spaces induced by the integrably admissible one-parameter groups.
