  We investigate the class of compact complex Hermitian-symplectic manifolds
$X$. For each Hermitian-symplectic metric $\omega$ on $X$, we introduce a
functional acting on the metrics in the Aeppli cohomology class of $\omega$ and
prove that its critical points (if any) must be K\"ahler when $X$ is
$3$-dimensional. We go on to exhibit these critical points as maximisers of the
volume of the metric in its Aeppli class and propose a Monge-Amp\`ere-type
equation to study their existence. Our functional is further utilised to define
a numerical invariant for any Aeppli cohomology class of Hermitian-symplectic
metrics that generalises the volume of a K\"ahler class. We obtain two
cohomological interpretations of this invariant. Meanwhile, we construct an
invariant in the form of an $E_2$-cohomology class, that we call the
$E_2$-torsion class, associated with every Aeppli class of Hermitian-symplectic
metrics and show that its vanishing is a necessary condition for the existence
of a K\"ahler metric in the given Hermitian-symplectic Aeppli class.
