  We consider the geometric and non-geometric faces of closed string vacua
arising by T-duality from principal torus bundles with constant H-flux and pay
attention to their double phase space description encompassing all toroidal
coordinates, momenta and their dual on equal footing. We construct a
star-product algebra on functions in phase space that is manifestly duality
invariant and substitutes for canonical quantization. The 3-cocycles of the
Abelian group of translations in double phase space are seen to account for
non-associativity of the star-product. We also provide alternative
cohomological descriptions of non-associativity and draw analogies with the
quantization of point-particles in the field of a Dirac monopole or other
distributions of magnetic charge. The magnetic field analogue of the R-flux
string model is provided by a constant uniform distribution of magnetic charge
in space and non-associativity manifests as breaking of angular symmetry. The
Poincare vector comes to rescue angular symmetry as well as associativity and
also allow for quantization in terms of operators and Hilbert space only in the
case of charged particles moving in the field of a single magnetic monopole.
