We present a new study of polarized elastic muon-electron scattering. The Born cross-section is calculated for arbitrary polarization of muon and electron. The complete photonic O(alpha) radiative corrections are determined for the case of longitudinally polarized muons and electrons. All calculations are done by two methods: semianalytic, which allows an implementation of the experimental cuts used for the analysis of MU-E scattering data from the beam polarimeter of the SMC experiment at CERN and completely analytic, which is used for cross checks The FORTRAN code MUELA realizes formulae of both approaches. We prove that certain experimental cuts lead to negligible radiative corrections in the muon beam polarization experiment.