  We expand the symmetry of the open finite-size SO(N) symmetric spin chain to
O(N). We partition its space of states into the eigenspaces of the parity
transformations in the flavor space, generating the subgroup
$Z_2^{\times(N-1)}$. It is proven that the lowest-energy states in these
eigenspaces are nondegenerate and assemble in antisymmetric tensors or
pseudotensors. At the valence-bond solid point, they constitute the
$2^{N-1}$-fold degenerate ground state with fully broken parity-transformation
symmetry.
