Neutrino oscillation data strongly support µ−τ symmetry as a good approximate flavor symmetry of the neutrino sector, which has to appear in any viable theory for neutrino mass-generation. The µ−τ breaking is not only small, but also the source of Dirac CP-violation. We conjecture that both discrete µ−τ 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 µ−τ breaking at low energies, leading to quantitative correlations between the apparently two small deviations θ 23 − 45 • and θ 13 − 0 • . This nontrivially connects the on-going measurements of mixing angle θ 23 with the upcoming experimental probes of θ 13 . We find that any deviation of θ 23 − 45 • must put a lower limit on θ 13 . Furthermore, we deduce the low energy Dirac and Majorana CP violations from a common soft-breaking phase associated with µ−τ 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 θ 13 mixing angle,• . In addition, we reveal a new hidden symmetry that dictates the solar mixing angle θ 12 by its group-parameter, and includes the conventional tri-bimaximal mixing as a special case, allowing deviations from it.

Existence of a mirror world in the universe is a fundamental way to restore the observed parity violation in weak interactions and provides the lightest mirror nucleon as a unique GeV-scale dark matter particle candidate. The visible and mirror worlds share the same spacetime of the universe and are connected by a unique space-inversion symmetry -the mirror parity (P ). We conjecture that the mirror parity is respected by the fundamental interaction Lagrangian, and study its spontaneous breaking from minimizing the Higgs vacuum potential. The domain wall problem is resolved by a unique soft breaking linear-term from the P -odd weak-singlet Higgs field. We also derive constraint from the Big-Bang nucleosynthesis. We then analyze the neutrino seesaw for both visible and mirror worlds, and demonstrate that the desired amounts of visible matter and mirror dark matter in the universe arise from a common origin of CP violation in the neutrino sector via leptogenesis. We derive the Higgs mass-spectrum and Higgs couplings with gauge bosons and fermions. We show their consistency with the direct Higgs searches and the indirect precision constraints. We further study the distinctive signatures of the predicted non-standard Higgs bosons at the LHC. Finally, we analyze the direct detections of GeV-scale mirror dark matter by TEXONO and CDEX experiments.