The recently developed diffraction technique of analytical investigation of the Rayleigh backscattering produced by an incident fundamental mode in a multimode optical fiber with an arbitrary refractive index profile is generalized to admit an arbitrary incident mode, either radial or azimuthal. The relative powers of all backscattered modes are determined with explicit formulas via the properly normalized transverse distributions of the incident and backscattered mode fields within the fiber cross section. The regularities conditioned by azimuthal indices are expressed via a universal set of coefficients, and dependences on the radial mode indices are estimated numerically. Excitation coefficients are shown to be symmetrical for any pair of incident and backscattered modes.