  The spectral properties of the post-merger gravitational-wave signal from a
binary of neutron stars encode a variety of information about the features of
the system and of the equation of state describing matter around and above
nuclear saturation density. Characterising the properties of such a signal is
an ``old'' problem, which first emerged when a number of frequencies were shown
to be related to the properties of the binary through ``quasi-universal''
relations. Here we take a new look at this old problem by computing the
properties of the signal in terms of the Weyl scalar $\psi_4$. In this way, and
using a database of more than 100 simulations, we provide the first evidence
for a new instantaneous frequency, $f^{\psi_4}_0$, associated with the instant
of quasi time-symmetry in the postmerger dynamics, and which also follows a
quasi-universal relation. We also derive a new quasi-universal relation for the
merger frequency $f^{h}_{\rm mer}$, which provides a description of the data
that is four times more accurate than previous expressions while requiring
fewer fitting coefficients. Finally, consistently with the findings of numerous
studies before ours, and using an enlarged ensamble of binary systems we point
out that the $\ell=2, m=1$ gravitational-wave mode could become comparable with
the traditional $\ell=2, m=2$ mode on sufficiently long timescales, with strain
amplitudes in a ratio $|h^{21}|/|h^{22}| \sim 0.1-1$ under generic orientations
of the binary, which could be measured by present detectors for signals with
large signal-to-noise ratio or by third-generation detectors for generic
signals should no collapse occur.
