AbstractThe oscillation of neutron n into mirror neutron $$n'$$ n ′ , its mass degenerate partner from dark mirror sector, can gradually transform the neutron stars into the mixed stars consisting in part of mirror dark matter. In quark stars $$n-n'$$ n - n ′ transitions are suppressed. We study the structure of mixed stars and derive the mass-radius scaling relations between the configurations of purely neutron star and maximally mixed star (MMS) containing equal amounts of ordinary and mirror components. In particular, we show that the MMS masses can be at most $$M^{\mathrm{max}}_{NS}/\sqrt{2}$$ M NS max / 2 , where $$M^\mathrm{max}_{NS}$$ M NS max is a maximum mass of a pure neutron star allowed by a given equation of state. We evaluate $$n-n'$$ n - n ′ transition rate in neutron stars, and show that various astrophysical limits on pulsar properties exclude the transition times in a wide range $$10^{5}\,\text {year}< \tau _\varepsilon < 10^{15}\,\text {year}$$ 10 5 year < τ ε < 10 15 year . For short transition times, $$\tau _\varepsilon < 10^5$$ τ ε < 10 5  year, the different mixed stars of the same mass can have different radii, depending on their age, which possibility can be tested by the NICER measurements. We also discuss subtleties related with the possible existence of mixed quark stars, and possible implications for the gravitational waves from the neutron star mergers and associated electromagnetic signals.