Can anyone explain why at a conical intersection the wavefunction changes sign? My understanding is that it is a test to see if the crossing IS indeed a conical intersection or just a coincidence of potential energy surfaces, which do not exhibit such phase changes upon periodic cycling. If this is correct ... why? Further what can we gain from knowing that?
A rather quick googling shows up that there is some Longuet-Higgins theorem, which states almost what you are looking for. For instance, the abstract of this paper says:
It is proved that if the wave function of a given electronic state changes sign when transported adiabatically round a loop in nuclear configuration space, then the state must become degenerate with another one at some point within the loop.
In the paper itself it is further stated that this in turn implies that
[the corresponding] potential energy surface intersects one of another electronic state
and that the theorem
makes it possible to diagnose the presence of an intersection between potential energy surfaces from the behaviour of the electronic wave function as it is transported adiabatically round a closed loop remote from the intersection
which answers both your questions, if I understand them correctly.