We discuss the representation of the SO(3) group by two-qubit maximally entangled states (MES). We analyse the correspondence between SO(3) and the set of two-qubit MES which are experimentally realizable. As a result, we offer a new interpretation of some recently proposed experiments based on MES. Employing the tools of quantum optics, we treat in terms of two-qubit MES some classical experiments in neutron interferometry, which showed the π-phase accrued by a spin-1/2 particle precessing in a magnetic field. By so doing, we can analyse the extent to which the recently proposed experiments - and future ones of the same sort - would involve essentially new physical aspects as compared with those performed in the past. We argue that the proposed experiments do extend the possibilities for displaying the double connectedness of SO(3), although for that to be the case it becomes necessary to map elements of SU(2) onto physical operations acting on two-level systems. © 2005 IOP Publishing Ltd.