  We investigate the invariance properties of general wavelet coorbit spaces
and Besov-type decomposition spaces under dilations by matrices. We show that
these matrices can be characterized by quasi-isometry properties with respect
to a certain metric in frequency domain. We formulate versions of this
phenomenon both for the decomposition and coorbit space settings.
  We then apply the general results to a particular class of dilation groups,
the so-called shearlet dilation groups. We present a general, algebraic
characterization of matrices the are coorbit compatible with a given shearlet
dilation group. We determine the groups of compatible dilations for a variety
of concrete examples.
