One of the fundamental elements for the construction of fermionic theories is the existence and choice of the spin structure. It is well known that the spectrum of the Dirac operator may depend on this choice. We show that a similar effect happens in Noncommutative Geometry. In the case study of the noncommutative torus, we prove the existence of inequivalent equivariant real spectral geometries that in the classical limit correspond to different choices of a spin structure. We show also the commutative limit of spectral geometries for two-dimensional spheres, which have no classical counterparts in differential geometry and belong to different topological classes.