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The periodic OSp(1|2) quantum spin chain has both a graded and a non-graded version. Naively, the Bethe ansatz solution for the non-graded version does not account for the complete spectrum of the transfer matrix, and we propose a simple mechanism for achieving completeness. In contrast, for the graded version, this issue does not arise. We also clarify the symmetries of both versions of the model, and we show how these symmetries are manifested in the degeneracies and multiplicities of the transfer-matrix spectrum. While the graded version has OSp(1|2) symmetry, the non-graded version has only SU(2) symmetry. Moreover, we obtain conditions for selecting the physical singular solutions of the Bethe equations. This analysis solves a lasting controversy over signs in the Bethe equations.