  We refine recent local unitary entanglement classification for symmetric pure
states of $n$ qubits (that is, states invariant under permutations of qubits)
using local unitary stabilizer subgroups and Majorana configurations.
Stabilizer subgroups carry more entanglement distinguishing power than do the
stabilizer subalgebras used in our previous work. We extend to mixed states
recent results about local operations on pure symmetric states by showing that
if two symmetric density operators are equivalent by a local unitary operation,
then they are equivalent via a local unitary operation that is the {\em same}
in each qubit. A geometric consequence, used in our entanglement
classification, is that two symmetric pure states are local unitary equivalent
if and only if their Majorana configurations can be interchanged by a rotation
of the Bloch sphere.
