  One of the hallmark of topological insulators is having conductivity
properties that are unaffected by the possible presence of defects. So far, for
classical waves in time reversal invariant systems, all attempts to obtain
topological modes have not displayed strict immunity to backscattering. Here,
we obtain exact perfect transmission across defects or disorder using a
combination of chiral and mirror symmetry. To demonstrate the principle, we
focus on a simple hexagonal lattice model with Kekule distortion displaying
topological edge waves and we show analytically and numerically that the
transmission across symmetry preserving defects is unity. Importantly, not only
the transmission is perfect, but no phase shift is induced making the defect
invisible. Our results rely on a generic lattice model, applicable to various
classical wave systems, which we realize to a high accuracy in an acoustic
system, and confirm the perfect transmission with numerical experiments. Our
work opens the door to new types of topological metamaterials exploiting
distinct symmetries to achieve robustness against defects or disorder. We
foresee that the versatility of our model will trigger new experiments to
observe topological perfect transmission and invisibility in various wave
systems, such as photonics, cold atoms or elastic waves.
