We have measured the azimuthal anisotropy of π⁰ production for 1<p(T)<18 GeV/c for Au+Au collisions at sqrt((s)NN)=200 GeV. The observed anisotropy shows a gradual decrease for 3≲p(T)≲7-10 GeV/c, but remains positive beyond 10 GeV/c. The magnitude of this anisotropy is underpredicted, up to at least ∼10 GeV/c, by current perturbative QCD (PQCD) energy-loss model calculations. An estimate of the increase in anisotropy expected from initial-geometry modification due to gluon saturation effects and fluctuations is insufficient to account for this discrepancy. Calculations that implement a path-length dependence steeper than what is implied by current PQCD energy-loss models show reasonable agreement with the data.