Time-odd (momentum) coupling between the nuclear and electronic wavefunctions in Jahn-Teller systems, although smaller than the traditional time-even (electrostatic) coupling terms, can dramatically change the geometric phase. In the cubic system, whose geometrical phase is a multiple of , time-odd coupling of modes with symmetry gives a three-dimensional parameter space and arbitrary geometrical phases. We show that for equal coupling the geometric vector potential takes the form of a monopole and hence that (as in the time-even case) the nuclear angular momentum for the system is shifted from integer to half-integer values.