The mantle-inner core gravitational (MICG) mode is the free mode axial oscillation between the mantle and inner core sustained by the gravitational torque between their degree 2 order 2 density structures.  As part of this mode, the tangent cylinder (TC) is entrained to move jointly with the inner core, and the oscillations of the TC launch Alfvén waves propagating in the region outside the TC.  Here, we investigate how the MICG is altered by the strength of the internal magnetic field in the core which controls the travelling speed of Alfvén waves.  We show that the MICG mode remains a distinct normal mode of oscillation of the core-mantle system only when Alfvén waves are attenuated before they traverse the width of the fluid core.  For an internal magnetic field strength of a few mT, as we expect in Earth's core, Alfvén waves can readily traverse the width of the core, and the MICG mode is absorbed into the spectrum of torsional oscillation (TO) modes.  The MICG period retains a dynamical influence, acting as a point of resonance for TO modes, and marking the transition from a TO mode that is weakly impacted by gravitational coupling to one in which the motion of the TC (including the inner core) is strongly restricted. Our results imply that the observed 6-year periodic signal in the length of day cannot be interpreted as the signature of the MICG mode and must instead be caused by TO modes, or more generally, by the propagation of Alfvén waves.