  Many experimental techniques aim at determining the Hamiltonian of a given
system. The Hamiltonian describes the system's evolution in the absence of
dissipation, and is often central to control or interpret an experiment. Here,
we theoretically propose and experimentally demonstrate a method for
Hamiltonian reconstruction from measurements over a large area of phase space,
overcoming the main limitation of previous techniques. A crucial ingredient for
our method is the presence of dissipation, which enables sampling of the
Hamiltonian through ringdown-type measurements. We apply the method to a
driven-dissipative system -- a parametric oscillator -- observed in a rotating
frame, and reconstruct the (quasi-)Hamiltonian of the system. Furthermore, we
demonstrate that our method provides direct experimental access to the
so-called symplectic norm of the stationary states of the system, which is tied
to the particle- or hole-like nature of excitations of these states. In this
way, we establish a method to unveil qualitative differences between the
fluctuations around stabilized minima and maxima of the nonlinear
out-of-equilibrium stationary states. Our method constitutes a versatile
approach to characterize a wide class of driven-dissipative systems.
