  In this paper we further develop the fluctuating hydrodynamics proposed in
arXiv:1511.03646 in a number of ways. We first work out in detail the classical
limit of the hydrodynamical action, which exhibits many simplifications. In
particular, this enables a transparent formulation of the action in physical
spacetime in the presence of arbitrary external fields. It also helps to
clarify issues related to field redefinitions and frame choices. We then
propose that the action is invariant under a $Z_2$ symmetry to which we refer
as the dynamical KMS symmetry. The dynamical KMS symmetry is physically
equivalent to the previously proposed local KMS condition in the classical
limit, but is more convenient to implement and more general. It is applicable
to any states in local equilibrium rather than just thermal density matrix
perturbed by external background fields. Finally we elaborate the formulation
for a conformal fluid, which contains some new features, and work out the
explicit form of the entropy current to second order in derivatives for a
neutral conformal fluid.
