We put forward a novel method for producing ultrarelativistic high-density high-polarization positrons through a single-shot interaction of a strong laser with a tilted solid foil. In our method, the driving laser ionizes the target, and the emitted electrons are accelerated and subsequently generate abundant γ photons via the nonlinear Compton scattering, dominated by the laser. These γ photons then generate polarized positrons via the nonlinear Breit-Wheeler process, dominated by a strong self-generated quasistatic magnetic field B^{S}. We find that placing the foil at an appropriate angle can result in a directional orientation of B^{S}, thereby polarizing positrons. Manipulating the laser polarization direction can control the angle between the γ photon polarization and B^{S}, significantly enhancing the positron polarization degree. Our spin-resolved quantum electrodynamics particle-in-cell simulations demonstrate that employing a laser with a peak intensity of about 10^{23} W/cm^{2} can obtain dense (≳10^{18} cm^{-3}) polarized positrons with an average polarization degree of about 70% and a yield of above 0.1 nC per shot. Moreover, our method is feasible using currently available or upcoming laser facilities and robust with respect to the laser and target parameters. Such high-density high-polarization positrons hold great significance in laboratory astrophysics, high-energy physics, and new physics beyond the standard model.