The Jordan-Wigner transformation plays an important role in spin models. However, the nonlocality of the transformation implies that a periodic chain of N spins is not mapped to a periodic or an anti-periodic chain of lattice fermions. Since only the N − 1 bond is different, the effect is negligible for large systems, while it is significant for small systems. In this paper, it is interesting to find that a class of periodic spin chains can be exactly mapped to a periodic chain and an antiperiodic chain of lattice fermions without redundancy when the Jordan-Wigner transformation is implemented. For these systems, possible high degeneracy is found to appear in not only the ground state, but also the excitation states. Further, we take the one-dimensional compass model and a new XY-XY model (σxσy − σxσy) as examples to demonstrate our proposition. Except for the well-known one-dimensional compass model, we will see that in the XY-XY model, the degeneracy also grows exponentially with the number of sites.