We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S.?Yokura and was proved by A.?Parusi¨½ski and P.?Pragacz. It also generalizes a formula of J.?Seade and T.?Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.