  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.
