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.