We show that Poincaré polarization singularities, spiraling like a tornado, can be generated by superimposing two orthogonally polarized, abruptly auto-focusing ring-Airy beams that carry orbital angular momentum (OAM). Seeded by phase vortices of the same helicity, which are adapted to the high-intensity rings of one of the superimposing beams, these polarization singularities follow trajectories that twist and shrink in an accelerating fashion along their propagation. Reaching angular acceleration that exceeds 120 rad/mm2, these Poincaré tornados can find application in singular optics, wavefront shaping, polarization engineering, and imaging through complex media.