The apodizing phase plate (APP) is a solid-state pupil optic that clears out a D-shaped area next to the core of the ensuing PSF. To make the APP more efficient for high-contrast imaging, its bandwidth should be as large as possible, and the location of the D-shaped area should be easily swapped to the other side of the PSF. We present the design of a broadband APP that yields two PSFs that have the opposite sides cleared out. Both properties are enabled by a half-wave liquid crystal layer, for which the local fast axis orientation over the pupil is forced to follow the required phase structure. For each of the two circular polarization states, the required phase apodization is thus obtained, and, moreover, the PSFs after a quarter-wave plate and a polarizing beam-splitter are complementary due to the antisymmetric nature of the phase apodization. The device can be achromatized in the same way as half-wave plates of the Pancharatnam type or by layering self-aligning twisted liquid crystals to form a monolithic film called a multi-twist retarder. As the VAPP introduces a known phase diversity between the two PSFs, they may be used directly for wavefront sensing. By applying an additional quarter-wave plate in front, the device also acts as a regular polarizing beam-splitter, which therefore furnishes high-contrast polarimetric imaging. If the PSF core is not saturated, the polarimetric dual-beam correction can also be applied to polarized circumstellar structure. The prototype results show the viability of the vector-APP concept.