We explore masses and mixings for Dirac neutrinos in models where lepton number is conserved, under the guidance of a hidden, but broken, $\mu-\tau$ exchange symmetry, that makes itself evident in the squared hermitian mass matrix. We study the parameter space in the most general theory as allowed by current neutrino oscillation experiment data... By using a general parameterization of the mass matrix which contains only observable parameters we stablish that the amount of breaking of the symmetry is in the range of the atmospheric mass scale, without regard to the neutrino hierarchy, the absolute neutrino mass and the Dirac CP phase. An estimate of the invisible branching ratio for a Higgs boson decaying into Dirac neutrinos, $H\rightarrow\nu\overline{\nu}$ , is given and compared to recent measurements in this context. (read more)