  We develop a formalism to discuss the properties of GENERIC systems in terms
of corresponding Hamiltonians that appear in the characterization of
large-deviation limits. We demonstrate how the GENERIC structure naturally
arises from a certain symmetry in the Hamiltonian, which extends earlier work
that has connected the large-deviation behaviour of reversible stochastic
processes to the gradient-flow structure of their deterministic limit. Natural
examples of application include particle systems with inertia.
