A thin disk-shaped container filled with a ferrofluid and suspended in a horizontal linearly polarized ac magnetic field can perform torsional vibrations around its vertical diameter. In contrast to a recently studied spherical pendulum, the cell is sensitive to the field direction: It exposes its edge to the stationary or slowly oscillating magnetic field, and its broad side to the field of a high frequency. When the amplitude of the latter field is increased, the state of rest gets destabilized, yielding to oscillations near the equilibrium. Further growth of the field strength results in the onset of the cell rotation. We describe sequences of local and global bifurcations which accompany those transitions.