The suppression by a magnetic field of the anomalous H=0 conducting phase in high-mobility silicon MOSFETs is independent of the angle between the field and the plane of the 2D electron system. In the presence of a parallel field large enough to fully quench the anomalous conducting phase, the system becomes similar to disordered GaAs/AlGaAs heterostructures: it is strongly insulating in zero (perpendicular) field, and exhibits reentrant insulator-quantum Hall effect-insulator transitions as a function of perpendicular field. The results demonstrate that the suppression of the low-$T$ phase is related only to the electrons' spin.