Spin geometry of the rational noncommutative torus

We discuss the structure of topologically non-trivial almost-commutative manifold for spectral triples realized on the algebra of smooth functions on the noncommutative torus with rational parameter. This is done by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of algebras, and with a spectral triple on a certain invariant subalgebra of the product algebra. The isomorphisms intertwine also the grading and real structure. This holds for all four inequivalent spin structures, which are explicitly constructed in terms of double coverings of the noncommutative torus (with arbitrary real parameter). These results are extended also to a class of curved (non flat)spectral triples, obtained as a perturbation of the standard one by eight central elements.