This talk will describe how the idea of paraparticles — a mathematically natural form of particle symmetry in quantum theory (albeit less familiar then Bose-Einstein and Fermi-Dirac) — adds to a recent debate between John Stachel and Oliver Pooley about whether general relativity and quantum theory provide analogous motivations for a structuralist ontology. In recent philosophy of physics, structuralism has suffered mixed fortunes. The Leibniz equivalence, a moderately structuralist position about spacetime points, has been widely accepted as the best interpretative option for general relativity, thanks in large part to Einstein's hole argument. On the other hand, anti-haecceitism (also known as the non-individuals view), as applied to quantum particles, is by no means the interpreters' favourite. My talk will begin with an incident in the recent history of these two structuralist positions. The story's central character is anti-haecceitism in quantum mechanics, with Leibniz equivalence in supporting role. The story begins with Stachel's attempt in 2002 to unify Leibniz equivalence and anti-haecceitism, by giving a single, overarching motivation for them both. This was his "generalized hole argument for sets". The argument is abductive: structuralism best explains, or makes palatable, the permutability of theories. In 2006, Pooley gave a reason to sharply divide the two structuralist positions (and some corrections to Stachel's argument). General relativistic models are typically non-symmetric, and so altered by the action of a permutation of spacetime points; but in quantum theory, the symmetrization postulate makes models (i.e. quantum states) invariant under permutations — so that their permutability stands in no need of special interpretation. So while the abductive argument for Leibniz equivalence may stand, there is no good corresponding argument for anti-haecceitism. (I will give short shrift to the view that the symmetrization postulate already smuggles in anti-haecceitism.) Thus Pooley's position toward quantum mechanics harks back to Steven French's insistence, over the years, that the physics of quantum theory underdetermines the metaphysics of particle individuality. But the plot thickens when, following Messiah and Greenberg (1964), we retrench from the symmetrization postulate, endorsing instead its better-justified cousin, the indistinguishability postulate — and thereby accept the possibility of paraparticles. Now the relevant formal differences between general relativity and quantum mechanics, which Stachel missed and Pooley emphasized, disappear; so that abductive support for anti-haecceitism in QM is rehabilitated. Therefore, also rehabilitated are the affinities between Leibniz equivalence and anti-haecceitism which Stachel originally articulated. Like Stachel, I argue that Leibniz equivalence and anti-haecceitism have a unified motivation: namely, one should collapse mathematical representations that differ only by the non-structural information they contain — expressed in these two theories by coordinates and particle labels respectively — which are considered to be unphysical descriptive artefacts. This approach can itself be placed within a practice which is yet more general, and in physics jargon, goes by the name of gauge: namely, the collapse (quotienting, or formal identification) of representations, to eliminate descriptive distinctions without a physical difference.