  The two discrete generators of the full Lorentz group $O(1,3)$ in $4D$
spacetime are typically chosen to be parity inversion symmetry $P$ and time
reversal symmetry $T$, which are responsible for the four topologically
separate components of $O(1,3)$. Under general considerations of quantum field
theory (QFT) with internal degrees of freedom, mirror symmetry is a natural
extension of $P$, while $CP$ symmetry resembles $T$ in spacetime. In
particular, mirror symmetry is critical as it doubles the full Dirac fermion
representation in QFT and essentially introduces a new sector of mirror
particles. Its close connection to T-duality and Calabi-Yau mirror symmetry in
string theory is clarified. Extension beyond the Standard model can then be
constructed using both left- and right-handed heterotic strings guided by
mirror symmetry. Many important implications such as supersymmetry, chiral
anomalies, topological transitions, Higgs, neutrinos, and dark energy, are
discussed.
