Spin-polarized transport through bistable magnetic adatoms or single-molecule magnets (SMMs), which exhibit both uniaxial and transverse magnetic anisotropy, is considered theoretically. The main focus is on the impact of transverse anisotropy on transport characteristics and the adatom's or SMM's spin. In particular, we analyze the role of quantum tunneling of magnetization (QTM) in the mechanism of the current-induced spin switching, and show that the QTM phenomenon becomes revealed as resonant peaks in the average values of the molecule's spin and in the charge current. These features appear at some resonant fields and are observable when at least one of the electrodes is ferromagnetic.