The symmetry of self-duality (Savit 1980 "Rev. Mod. Phys. 52" 453) of some models of statistical mechanics and quantum field theory is discussed for finite spin blocks of the Ising chain in a transverse magnetic field. The existence of this symmetry in a specific type of these blocks, and not in others, is manifest by the degeneracy of their energy levels. This illustrates the fact that in quantum mechanics, more symmetry in the Hamiltonian induces more degeneracy in the energy spectrum. (Contains 5 figures.)