  We investigate the relation between Cartan decompositions of the unitary
group and discrete quantum symmetries. To every Cartan decomposition there
corresponds a quantum symmetry which is the identity when applied twice. As an
application, we describe a new and general method to obtain Cartan
decompositions of the unitary group of evolutions of multipartite systems from
Cartan decompositions on the single subsystems. The resulting decomposition,
which we call of the odd-even type, contains, as a special case, the
concurrence canonical decomposition (CCD) presented in the context of
entanglement theory. The CCD is therefore extended from the case of a
multipartite system of n qubits to the case where the component subsystems have
arbitrary dimension.
