We design a numerical scheme for transport equations with oscillatory periodic scattering coefficients. The scheme is asymptotic preserving in the diffusion limit as Knudsen number goes to zero. It also captures the homogenization limit as the length scale of the scattering coefficient goes to zero. The proposed method is based on the construction of multiscale finite element basis and a Galerkin projection based on the even-odd decomposition. The method is analyzed in the asymptotic regime, as well as validated numerically.