  The twined almost commutative structure of the standard spectral triple on
the noncommutative torus with rational parameter is exhibited, by showing
isomorphisms with a spectral triple on the algebra of sections of certain
bundle of algebras, and a spectral triple on a certain invariant subalgebra of
the product algebra. These 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.
