  In the canonical seesaw mechanism we require the relevant neutrino mass terms
to be invariant under the $S^{}_3$ charge-conjugation transformations of left-
and right-handed neutrino fields. Then both the Dirac mass matrix $M^{}_{\rm
D}$ and the right-handed neutrino mass matrix $M^{}_{\rm R}$ are well
constrained, so is the effective light Majorana neutrino mass matrix $M^{}_\nu$
via the seesaw formula. We find that these mass matrices can be classified into
22 categories, among which some textures respect the well-known $\mu$-$\tau$
permutation or reflection symmetry and flavor democracy. It is also found that
there exist remarkable structural equalities or similarities between $M^{}_\nu$
and $M^{}_{\rm R}$, reflecting a seesaw mirroring relationship between light
and heavy Majorana neutrinos. We calculate the corresponding light neutrino
masses and flavor mixing parameters as well as the CP-violating asymmetries in
decays of the lightest heavy Majorana neutrino, and show that only the flavored
leptogenesis mechanism is possible to work for three categories of $M^{}_{\rm
D}$ and $M^{}_{\rm R}$ in the $S^{}_3$ reflection symmetry limit.
