  Let $(X,\omega)$ be a compact hermitian manifold of dimension $n$. We study
the asymptotic behavior of Monge-Amp\`ere volumes $\int_X (\omega+dd^c
\varphi)^n$, when $\omega+dd^c \varphi$ varies in the set of hermitian forms
that are $dd^c$-cohomologous to $\omega$. We show that these Monge-Amp\`ere
volumes are uniformly bounded if $\omega$ is "strongly pluripositive", and that
they are uniformly positive if $\omega$ is "strongly plurinegative". This
motivates the study of the existence of such plurisigned hermitian metrics.
  We analyze several classes of examples (complex parallelisable manifolds,
twistor spaces, Vaisman manifolds) admitting such metrics, showing that they
cannot coexist. We take a close look at $6$-dimensional nilmanifolds which
admit a left-invariant complex structure, showing that each of them admit a
plurisigned hermitian metric, while only few of them admit a pluriclosed
metric. We also study $6$-dimensional solvmanifolds with trivial canonical
bundle.
