The interaction between laser and relativistic electron beams is a promising source of very energetic X rays. We present an accurate model for the collisions between very intense linearly polarized laser beams, corresponding to relativistic parameters of the order of unity or greater, and electrons having energies up to 100 MeV. Our approach uses only one approximation, namely it neglects the radiative corrections. We consider the two cases in which the laser field polarization is either perpendicular or parallel to the plane defined by the directions of propagation of the laser beam and electron beam, and calculate accurately the properties of the σ and π polarized scattered beams. The angle between the directions of the laser and electron beams, denoted by θL, is allowed to have arbitrary values, so that the widely analyzed 180° and 90° geometries, in which the two beams collide, respectively, head on and perpendicularly, are particular cases. We prove that the polarization properties of the scattered beam depend on the angle θL. By varying this angle, the polarization of the scattered beam can be varied between the two limit configurations in which the electromagnetic field of the scattered beam is σ or π polarized with respect to the scattering plane. Our theoretical results are in good agreement with experimental results published in literature. Our model shows that current technologies can be used to produce hard harmonics of the scattered radiations. These harmonics can have relatively high intensities comparable to the intensities of the first harmonics, and energies higher than 1 MeV. Our results lead to the possibility to realize an adjustable photon source with both the energy and polarization of the scattered radiations accurately controlled by the value of the θL angle.