  We study the interconversion of multipartite symmetric $N$-qubit states under
stochastic local operations and classical communication (SLOCC). We demonstrate
that if two symmetric states can be connected with a nonsymmetric invertible
local operation (ILO), then they belong necessarily to the separable, W, or GHZ
entanglement class, establishing a practical method of discriminating subsets
of entanglement classes. Furthermore, we prove that there always exists a
symmetric ILO connecting any pair of symmetric $N$-qubit states equivalent
under SLOCC, simplifying the requirements for experimental implementations of
local interconversion of those states.
