A previously unknown "spin-other-orbit" effect of photon modes is predicted by solving a Schrodinger-like equation for a non-relativistic electron interacting with a two-mode intense photon field propagating in two perpendicular directions. A multiphoton ionization experiment to test this effect is suggested. Transition rates which manifest this effect to be measured in the experiment are calculated and presented graphically. A detailed comparison between this effect and the well-known spin-other-orbit effect of electrons is included.