Cyclic phases at an n-fold degeneracy

We study the sets of possible sign changes that can occur when n real quant um states are transported adiabatically around an n-fold degeneracy. The al lowed sign changes include those predicted by the familiar spin j = (n - 1) /2 model as a subset, but we also find at least one other possible set of s ign changes for each n > 2. Experimentally verifiable quadrilateral billiar d examples are presented to illustrate these additional possibilities for 3 - and 4-fold degeneracies.