Using field effect devices with side gates, we modulate the 2-dimensional electron gas hosted at the LaAlO3/SrTiO3 interface to study the temperature and doping evolution of the magnetotransport. The analysis of the data reveals different transport regimes depending on the interplay between the different (elastic, inelastic, and spin-orbit) scattering times and their temperature dependencies. We find that the spin-orbit interaction strongly affects the low-temperature transport in the normal state in a very large region of the phase diagram, extending beyond the superconducting dome.