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Using the analytical expressions for the genuine eigenfunctions φμν(z) and eigenvalues Eμ,ν, of open, bounded and quasi-bounded finite periodic systems, we derive the eigenfunctions space-inversion symmetry relations. The superlattice eigenfunctions symmetries, closely related with the symmetries and zeros of the Chebyshev polynomials of the second kind Un, are fully written in terms of the number of unit cells n, the subband index μ and the intra-subband index ν.