  We investigate the relation between local unitary symmetries and entanglement
invariants of multi-qubit systems. The Hilbert space of such systems can be
stratified in terms of states with different types of symmetry. We review the
connection between this stratification and the ring of entanglement invariants
and the corresponding geometric description in terms of algebraic varieties. On
a stratum of a non-trivial symmetry group the invariants of the symmetry
preserving operations gives a sufficient description of entanglement. Finding
these invariants is often a simpler problem than finding the invariants of the
local unitary group. The conditions, as given by the Luna-Richardson theorem,
for when the ring of such invariants is isomorphic to the ring of local unitary
invariants on the stratum are discussed. As an example we consider symmetry
groups that can be diagonalized by local unitary operations and for which the
group action on each qubit is non-trivial. On the stratum of such a symmetry
the entanglement can be described in terms of a canonical form and the
invariants of the symmetry preserving operations. This canonical form and the
invariants are directly determined by the symmetry group. Further, we briefly
discuss how some recently proposed entanglement classification schemes capture
symmetry properties.
