  Recently, shearlet groups have received much attention in connection with
shearlet transforms applied for orientation sensitive image analysis and
restoration. The square integrable representations of the shearlet groups
provide not only the basis for the shearlet transforms but also for a very
natural definition of scales of smoothness spaces, called shearlet coorbit
spaces. The aim of this paper is twofold: first we discover isomorphisms
between shearlet groups and other well-known groups, namely extended Heisenberg
groups and subgroups of the symplectic group. Interestingly, the connected
shearlet group with positive dilations has an isomorphic copy in the symplectic
group, while this is not true for the full shearlet group with all nonzero
dilations. Indeed we prove the general result that there exist, up to adjoint
action of the symplectic group, only one embedding of the extended Heisenberg
algebra into the Lie algebra of the symplectic group.
  Having understood the various group isomorphisms it is natural to ask for the
relations between coorbit spaces of isomorphic groups with equivalent
representations. These connections are examined in the second part of the
paper. We describe how isomorphic groups with equivalent representations lead
to isomorphic coorbit spaces. In particular we apply this result to square
integrable representations of the connected shearlet groups and metaplectic
representations of subgroups of the symplectic group. This implies the
definition of metaplectic coorbit spaces.
  Besides the usual full and connected shearlet groups we also deal with
Toeplitz shearlet groups.
