In this work, we report the realization of broadband binary-reflection-phase metasurfaces that simultaneously exhibit undistorted transmission wavefront. Such a unique functionality is bestowed by leveraging mirror symmetry in the metasurface design. Under the normal incidence of waves polarized along the mirror surface, a broadband binary-phase pattern with π phase difference is induced in the cross-polarized reflection, while the co-polarized transmission and reflection are unaffected by the binary-phase pattern. Consequently, the cross-polarized reflection can be flexibly manipulated by designing the binary-phase pattern, without distorting the wavefront in transmission. The phenomena of reflected-beam splitting and undistorted transmission wavefront are hereby experimentally validated in a broad bandwidth from 8 GHz to 13 GHz. Our findings reveal a unique mechanism to realize independent manipulation of reflection with undistorted transmission wavefront in a broad spectrum, which has potential implications in meta-domes and reconfigurable intelligent surfaces.