The PHENIX experiment at the BNL Relativistic Heavy Ion Collider has measured second- and third-order Fourier coefficients of the azimuthal distributions of direct photons emitted at midrapidity in Au + Au collisions at root S-NN = 200 GeV for various collision centralities. Combining two different analysis techniques, results were obtained in the transverse momentum range of 0.4 < p(T) < 4.0 GeV/c. At low p(T) the second-order coefficients, nu(2), are similar to the ones observed in hadrons. Third-order coefficients, nu(3), are nonzero and almost independent of centrality. These new results on nu(2) and nu(3), combined with previously published results on yields, are compared to model calculations that provide yields and asymmetries in the same framework. Those models are challenged to explain simultaneously the observed large yield and large azimuthal anisotropies.