  We derive predictions for the Nadir angle (theta_n) dependence of the ratio
Nmu-/Nmu+ of the rates of the mu- and mu+ multi-GeV events, and for the mu- -
mu+ event rate asymmetry, A_{mu-mu+}=[Nmu- - Nmu+]/[Nmu- + Nmu+], in
iron-magnetized calorimeter detectors (MINOS, INO) in the case of 3-neutrino
oscillations of the atmospheric nu_mu and antinu_mu, driven by one neutrino
mass squared difference, |Delta m^2_{31}| >> Delta m^2_{21}. The asymmetry
A_{mu- mu+} (the ratio Nmu-/Nmu+) is shown to be particularly sensitive to the
Earth matter effects in the atmospheric neutrino oscillations, and thus to the
values of sin^2(theta13) and sin^2(theta23), theta13 and theta23 being the
neutrino mixing angles limited by the CHOOZ and Palo Verde experiments and that
responsible for the dominant atmospheric nu_mu -> nu_tau (antinu_mu ->
antinu_tau) oscillations. It is also very sensitive to the type of neutrino
mass spectrum which can be with normal (Delta m^2_{31}>0) or with inverted
(Delta m^2_{31} < 0) hierarchy. We find that for sin^2(theta23)>0.50, sin^2(2
theta13)>0.06 and Delta m^2_{31}=(2-3) 10^{-3} eV^2, the Earth matter effects
produce a relative difference between the integrated asymmetries barA_{mu- mu+}
and barA^{2nu}_{mu- mu+}$ in the mantle (cos(theta_n)=0.30-0.84) and core
(cos(theta_n)=0.84-1.0) bins, which is bigger in absolute value than ~15%, can
reach the values of (30-35)%, and thus can be sufficiently large to be
observable. The sign of the indicated asymmetry difference is anticorrelated
with the sign of Delta m^2_{31}. An observation of the Earth matter effects in
the Nadir angle distribution of the asymmetry A_{mu- mu+} (ratio Nmu-/Nmu+)
would clearly indicate that sin^2(2 theta13)>0.06 and sin^2(theta23)>0.50, and
would lead to the determination of the sign of Delta m^2_{31}.
