Recently, reconfigurable intelligent surfaces (RISs) gained notable consideration due to their ability to provide efficient and cost-effective wireless communication networks. However, this powerful concept often suffers from simplistic modeling which underestimates such features of RIS as the resonant frequency dispersion and strong angular dependency of the reflection phases for both TE and TM polarizations of the incident wave. The angular and polarization instability of the reflection phase is a fundamental restriction of RISs, especially restrictive if the operation frequency band is broad. In this paper, we address this challenge for a binary RIS performed as a metasurface. We have studied the reflection phase frequency dispersion (RPFD) analytically that allowed us to engineer the needed angular and polarization properties of the RIS. Our RIS is a self-resonant grid of Jerusalem crosses located on a thin metal-backed dielectric substrate. Adjacent crosses are connected by switchable capacitive loads. We have shown the advantage of our metasurface compared to switchable mushroom-field structures and meta-gratings of resonant patches. An RIS is also fabricated and measured, and the experimental results corroborate well our numerical full wave simulations and analytical predictions.INDEX TERMS Reconfigurable intelligent surface (RIS), wireless communication, angular stability.