In the framework of a left-right model containing mirror fermions (LRMM) with gauge group SU(3)C⊗SU(2)L⊗SU(2)R⊗U(1)Y′, we estimate the neutrino masses, which are found to be consistent with their experimental bounds and hierarchy. We also estimate the decay rates of heavy Majorana neutrinos in the channels N→W±l∓, N→Zνl and N→Hνl, which are roughly equal for large values of the heavy neutrino mass. Starting from the most general Majorana neutrino mass matrix, the smallness of active neutrino masses turns out from the interplay of the hierarchy of the involved scales and the double application of seesaw mechanism. An appropriate ansatz on the structure of the neutrino mass matrix imposing a symmetric mixing of electron neutrino with muon and tau neutrinos leads to tribimaximal mixing matrix for light neutrinos.