  We discuss the representation of the $SO(3)$ group by two-qubit maximally
entangled states (MES). We analyze 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 $\pi $-phase
accrued by a spin-$1/2$ particle precessing in a magnetic field. By so doing,
we can analyze 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 results necessary to map
elements of $SU(2)$ onto physical operations acting on two-level systems.
