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We analyze the class of nonlocal realistic theories that was originally considered by Leggett [Found. Phys. 33, 1469 (2003)] and tested by us in a recent experiment [Nature (London) 446, 871 (2007)]. We derive an incompatibility theorem that works for finite numbers of polarizer settings and that does not require the previously assumed rotational symmetry of the two-particle correlation functions. The experimentally measured case involves seven different measurement settings. Using polarization-entangled photon pairs, we exclude this broader class of nonlocal realistic models by experimentally violating a new Leggett-type inequality by 80 standard deviations.