  The effect of rotation on the formation of the magnetic dual chiral density
wave (\MD) in a dense and magnetized cold quark matter is studied. This phase
is supposed to exist in the extreme conditions prevailing, e.g., in a neutron
star. These conditions are, apart from high densities and strong magnetic
fields, a relatively large angular velocity. To answer the question of whether
the rotation enhances or suppresses the formation of this phase, we first
determine the effect of rotation on the energy dispersion relation of a
fermionic system in the presence of a constant magnetic field and then focus on
the thermodynamic potential of the model at low temperature $T$ and finite
chemical potential $\mu$. The thermodynamic potential consists, in particular,
of an anomalous part leading to certain topological effects. We show that in
comparison with the nonrotating case, a term proportional to the angular
velocity appears in this anomalous potential. We then solve the corresponding
gap equations to the chiral and spatial modulation condensates, and study the
dependence of these dynamical variables on the chemical potential ($\mu$),
magnetic field ($eB$), and angular velocity ($\Omega$). It turns out that the
interplay between these parameters suppresses the formation of the \MD~phase in
relevant regimes for cold neutron stars. This is interpreted as the
manifestation of the inverse magnetorotational catalysis, which is also
reflected in the phase portraits $eB$-$\mu$, $eB$-$\Omega R$, and $\mu$-$\Omega
R$, explored in this work.
