Ferroelectric switching in BiFeO₃ multiferroic thin films was studied by piezoresponse force microscopy, as a function of the tip voltage and sweep direction, for samples with two different intrinsic domain structures. In all films, the switched polarization direction follows the in-plane and out-of-plane components of the highly inhomogeneous electric field applied by the microscope tip. In films with 'bubble-like' intrinsic domains, we observed in-plane switching assisted by out-of-plane switching for lower voltage values, and independent in-plane and out-of-plane switching for higher voltages, in both cases allowing full control of the ferroelectric polarization depending on the tip voltage polarity and sweep direction. In films with 'stripe-like' intrinsic domains, independent in-plane and out-of-plane switching was observed, but unswitched stripe domains prevented full control of the ferroelectric polarization over large areas. We correlate the observed switching behavior with the field-driven onset of a highly distorted tetragonal phase predicted by ab initio calculations, which leads to a very high in-plane susceptibility during the return to the non-distorted monoclinic phase when the field is decreased. Depending on the specific strain and disorder present in the sample, the transition towards the highly distorted phase may be asymmetrized, and easier to reach when an electric field opposite to the out-of-plane polarization direction is applied.