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The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit states cannot be obtained from nor transformed to any state (i.e., they are isolated), which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that generic n-qubit symmetric states are also isolated, we consider particular families for which we can determine to be, on the contrary, endowed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. In the course of the study of LOCC transformations, we also characterize the local symmetries of symmetric states.