  The electromagnetic properties of an extremely thin metallic checkerboard
drastically change from resonant reflection (transmission) to resonant
transmission (reflection) when the local electrical conductivity at the
interconnection points of the checkerboard is switched. To date, such critical
transitions of metasurfaces have been applied only when they have 4-fold
rotational symmetry, and their application to polarization control, which
requires anisotropy, has been unexplored. To overcome this applicability
limitation and open up new pathways for dynamic deep-subwavelength polarization
control by utilizing critical transitions of checkerboard-like metasurfaces, we
introduce a universal class of anisotropic Babinet-invertible metasurfaces
enabling transmission-reflection switching for each orthogonally polarized
wave. As an application of anisotropic Babinet-invertible metasurfaces, we
experimentally realize a reconfigurable terahertz polarizer whose transmitting
axis can be dynamically rotated by $90^\circ$.
