  Hydrodynamics is a powerful emergent theory for the large-scale behaviours in
many-body systems, quantum or classical. It is a gradient series expansion,
where different orders of spatial derivatives provide an effective description
on different length scales. We here report the first general derivation of
third-order, or "dispersive", terms in the hydrodynamic expansion. We obtain
fully general Kubo-like expressions for the associated hydrodynamic
coefficients, and we determine their expressions in quantum integrable models,
introducing in this way purely quantum higher-order terms into generalised
hydrodynamics. We emphasise the importance of hydrodynamic gauge fixing at
diffusive order, where we claim that it is parity-time-reversal, and not
time-reversal, invariance that is at the source of Einstein's relation,
Onsager's reciprocal relations, the Kubo formula and entropy production. At
higher hydrodynamic orders we introduce a more general, n-th order "symmetric"
gauge, which we show implies the validity of the higher-order hydrodynamic
description.
