  We present an inertia-augmented relaxed micromorphic model that enriches the
relaxed micromorphic model previously introduced by the authors via a term
$\text{Curl}\dot{P}$ in the kinetic energy density. This enriched model allows
us to obtain a good overall fitting of the dispersion curves while introducing
the new possibility of describing modes with negative group velocity that are
known to trigger negative refraction effects. The inertia-augmented model also
allows for more freedom on the values of the asymptotes corresponding to the
cut-offs. In the previous version of the relaxed micromorphic model, the
asymptote of one curve (pressure or shear) is always bounded by the cut-off of
the following curve of the same type. This constraint does not hold anymore in
the enhanced version of the model. While the obtained curves' fitting is of
good quality overall, a perfect quantitative agreement must still be reached
for very small wavelengths that are close to the size of the unit cell.
