  We study the backward scatterings of plane waves by reciprocal scatterers and
reveal that $n$-fold ($n\geq3$) rotation symmetry is sufficient to secure
invariant backscattering for arbitrarily-polarized incident plane waves. It is
further demonstrated that the same principle is also applicable for infinite
periodic structures in terms of reflection, which simultaneously guarantees the
transmission invariance if there are neither Ohmic losses nor extra diffraction
channels. At the presence of losses, extra reflection symmetries (with
reflection planes either parallel or perpendicular to the incident direction)
can be incorporated to ensure simultaneously the invariance of transmission and
absorption. The principles we have revealed are protected by fundamental laws
of reciprocity and parity conservation, which are fully independent of the
optical or geometric parameters of the photonic structures. The optical
invariance obtained is intrinsically robust against perturbations that preserve
reciprocity and the geometric symmetries, which could be widely employed for
photonic applications that require stable backscatterings or reflections.
