  We investigate the role played by symmetries in the perturbative expansion of
the large-scale structure. In particular, we establish which of the
coefficients of the perturbation theory kernels are dictated by symmetries and
which not. Up to third order in perturbations, for the dark matter density
contrast (and for the dark matter velocity) only three coefficients are not
fixed by symmetries and depend on the particular cosmology. For generic biased
tracers, where number/mass and momentum conservation cannot be imposed in
general, this number rises to seven in agreement with other bias expansions
discussed in the literature. A crucial role in our analysis is provided by
extended Galilean invariance, which follows from diffeomorphism invariance in
the non-relativistic limit. We identify a full hierarchy of extended Galilean
invariance constraints, which fix the analytic structure of the perturbation
theory kernels as the sums of an increasing number of external momenta vanish.
Our approach is especially relevant for non-standard models that respect the
same symmetries as $\Lambda$CDM and where perturbation theory at higher orders
has not been exhaustively explored, such as dark energy and modified gravity
scenarios. In this context, our results can be used to systematically extend
the bias expansion to higher orders and set up model independent analyses.
