Photodissociation of water at a wavelength of 121.6 nanometers has been investigated by using the H-atom Rydberg tagging technique. A striking even-odd intensity oscillation was observed in the OH(X) product rotational distribution. Model calculations attribute this oscillation to an unusual dynamical interference brought about by two dissociation pathways that pass through dissimilar conical intersections of potential energy surfaces, but result in the same products. The interference pattern and the OH product rotational distribution are sensitive to the positions and energies of the conical intersections, one with the atoms collinear as H-OH and the other as H-HO. An accurate simulation of the observations would provide a detailed test of global H(2)O potential energy surfaces for the three (&Xtilde;/A/&Btilde;) contributing states. The interference observed from the two conical intersection pathways provides a chemical analog of Young's well-known double-slit experiment.