  Symmetry plays fundamental role in physics and the nature of symmetry changes
in non-Hermitian physics. Here the symmetry-protected scattering in
non-Hermitian linear systems is investigated by employing the discrete
symmetries that classify the random matrices. The even-parity symmetries impose
strict constraints on the scattering coefficients: the time-reversal (C and K)
symmetries protect the symmetric transmission or reflection; the
pseudo-Hermiticity (Q symmetry) or the inversion (P) symmetry protects the
symmetric transmission and reflection. For the inversion-combined time-reversal
symmetries, the symmetric features on the transmission and reflection
interchange. The odd-parity symmetries including the particle-hole symmetry,
chiral symmetry, and sublattice symmetry cannot ensure the scattering to be
symmetric. These guiding principles are valid for both Hermitian and
non-Hermitian linear systems. Our findings provide fundamental insights into
symmetry and scattering ranging from condensed matter physics to quantum
physics and optics.
