We show that there exist six parallel textures of the charged lepton and neutrino mass matrices with six vanishing entries, whose phenomenological consequences are exactly the same. These isomeric lepton mass matrices are compatible with current experimental data at the 3σ level. If the seesaw mechanism and the Fukugita-Tanimoto-Yanagida hypothesis are taken into account, it will be possible to fit the experimental data at or below the 2σ level. In particular, the maximal atmospheric neutrino mixing can be reconciled with a strong neutrino mass hierarchy in the seesaw case.