In many physical theories, a permutation on objects naturally induces a permutation on states, i.e. on the mathematical representatives of physical states of affairs. In general, the induced permutation is not trivial, i.e. is not the identity map. For example, in quantum theory: although a permutation of indistinguishable bosons induces only the identity map on the (symmetric) states, for indistinguishable fermions, an odd permutation induces a sign-change in the vector-state. And for paraparticles, the permuted state (the permutation-image of a given vector-state) in general does not even lie in the same ray as the given state. Non-trivial permutations of states raise the question: do the permuted state and the given one (the image and the argument) represent the same physical states of affairs? In physicists' jargon: are their differences as mathematical objects examples of gauge freedom? We answer Yes to this question — a position we call `structuralism': hence our title. This paper develops this answer, for indistinguishable particles in quantum theory (with a fixed number of particles). We first discuss how this answer fits with our advocacy (elsewhere) of a view we call QII (for `qualitative individuality with indiscernibles'). We emphasise paraparticles, so as to best illustrate the merits of structuralism. Since paraparticles have, unfortunately, been largely ignored in the philosophy of physics literature, we give full details for the case of three particles ($S_3$). We also stress, pace the usual dismissive comment that paraparticles do not occur in nature, that paraparticle states do occur in the orthodox (non-field) theory of quarks (albeit only as factor states in a state that as a whole is anti-symmetrized). This is sufficient for our philosophical morals: nothing turns on the fact that the composite state for all degrees of freedom is always anti-symmetrized.