  Neutrino oscillation data strongly support mu-tau symmetry as a good
approximate flavor symmetry of the neutrino sector, which has to appear in any
viable theory for neutrino mass-generation. The mu-tau breaking is not only
small, but also the source of Dirac CP-violation. We conjecture that both
discrete mu-tau and CP symmetries are fundamental symmetries of the seesaw
Lagrangian (respected by interaction terms), and they are only softly broken,
arising from a common origin via a unique dimension-3 Majorana mass-term of the
heavy right-handed neutrinos. From this conceptually attractive and simple
construction, we can predict the soft mu-tau breaking at low energies, leading
to quantitative correlations between the apparently two small deviations
\theta_{23} - 45^o and \theta_{13} - 0^o. This nontrivially connects the
on-going measurements of mixing angle \theta_{23} with the upcoming
experimental probes of \theta_{13}. We find that any deviation of \theta_{23} -
45^o must put a lower limit on \theta_{13}. Furthermore, we deduce the low
energy Dirac and Majorana CP violations from a common soft-breaking phase
associated with mu-tau breaking in the neutrino seesaw. Finally, from the soft
CP breaking in neutrino seesaw we derive the cosmological CP violation for the
baryon asymmetry via leptogenesis. We fully reconstruct the leptogenesis
CP-asymmetry from the low energy Dirac CP phase and establish a direct link
between the cosmological CP-violation and the low energy Jarlskog invariant. We
predict new lower and upper bounds on the \theta_{13} mixing angle, 1^o <
\theta_{13} < 6^o. In addition, we reveal a new hidden symmetry that dictates
the solar mixing angle \theta_12 by its group-parameter, and includes the
conventional tri-bimaximal mixing as a special case, allowing deviations from
it.
