Two decades ago, it was predicted [Y. S. Kim et al., Phys. Rev. Lett. 46, 1326 (1981)] that relativistic effects should alter the dynamics of the photoionization process in the vicinity of Cooper minima. The present experimental and theoretical study of the angular distributions of Xe 4d3/2 and 4d5/2 photoelectrons demonstrates this effect for the first time. The results clearly imply that relativistic effects are likely to be important for intermediate- Z atoms at most energies