  It has been shown in [32,33] in the framework of Nambu--Jona-Lasinio model
with the assumption of spatially homogeneous condensates that in the
large-$N_{c}$ limit ($N_{c}$ is the number of quark colours) there exist three
dual symmetries of the thermodynamic potential, which describes dense quark
matter with chiral and isospin imbalances. The main duality is between the
chiral symmetry breaking and the charged pion condensation phenomena. There
have been a lot of studies and hints that the ground state could be
characterized by spatially inhomogeneous condensates, so the question arises if
duality is a rather deep property of the phase structure or just accidental
property in the homogeneous case. In this paper we have shown that even if the
phase diagram contains phases with spatially inhomogeneous condensates, it
still possesses the property of this main duality. Two other dual symmetries
are not realized in the theory if it is investigated within an inhomogeneous
approach to a ground state. From various earlier studied aspects of QCD phase
diagram of isospin asymmetric matter with possible inhomogeneous condensates,
in the present paper the unified picture and full phase diagram of isospin
imbalanced dense quark matter have been assembled. Acting on this diagram by a
dual transformation, we obtained, in the framework of an approach with
spatially inhomogeneous condensates and without any calculations, a full phase
diagram of chirally asymmetric dense medium. This example shows that the
duality is not just entertaining mathematical property but an instrument with
very high predictivity power. The obtained phase diagram is quite rich and
contains various spatially inhomogeneous phases.
