Dielectric metasurfaces composed of subwavelength resonators are widely employed for manipulating electromagnetic waves over a broad frequency spectrum ranging from microwaves to optics. Here, a novel type of metasurfaces, created by a periodic lattice of elliptical holes fabricated in a thin dielectric membrane, is studied both theoretically and experimentally. Such membrane metasurfaces demonstrate polarization-selective behavior, and they change their specific functionality from a reflective magnetic mirror, for one polarization, to a transparent Huygens' surface, for the orthogonal polarization. Such a polarization dependence is achieved by manipulating the interference of Mie-resonant multipoles through optimizing the design of the elliptic holes in the dielectric membrane. Such membrane metasurfaces can be employed as flat components with polarization-multiplexed functionalities, bringing benefits for the integration of ultracompact signal manipulation systems. © 2020 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim