  Type I and type II seesaw contributions to the mass matrix of light neutrinos
are inherently related if left-right symmetry is realized at high energy
scales. We investigate implications of such a relation for the interpretation
of neutrino data. We proved recently that the left-right symmetric seesaw
equation has eight solutions, related by a duality property, for the mass
matrix of right-handed neutrinos $M_R$. In this paper the eight allowed
structures of $M_R$ are reconstructed analytically and analyzed numerically in
a bottom-up approach. We study the dependence of right-handed neutrino masses
on the mass spectrum of light neutrinos, mixing angle $\theta_{13}$, leptonic
CP violation, scale of left-right symmetry breaking and on the hierarchy in
neutrino Yukawa couplings. The structure of the seesaw formula in several
specific SO(10) models is explored in the light of the duality. The outcome of
leptogenesis may depend crucially on the choice among the allowed structures of
$M_R$ and on the level crossing between right-handed neutrino masses.
