For a large class of one-dimensional spin-1 Hamiltonians with open boundary conditions, we show that there is a unitary transformation for which the off-diagonal matrix elements of the transformed Hamiltonian are non-positive. We use this to show that the ground state of a finite chain is at most fourfold degenerate, and that the expectation of the string observable of den Nijs and Rommelse in the ground state is bounded below by the expectation of the usual Neel order parameter. (This was proved for a smaller class of Hamiltonians by Kennedy and Tasaki.) The class of Hamiltonians to which our results apply include the general isotropic Hamiltonian Sigma _i(S_i.S_i+1- beta (Si.S_i+1)²) for beta >-1. For the usual Heisenberg Hamiltonian the transformed Hamiltonian is - Sigma _iT_i.T_i+1 where the operators T=(T^x, T^y, T^z) satisfy anticommutation relations like (T^x, T^y)=T^z. We can also use this transformation to obtain variational bounds on the ground-state energy. The transformation used here is closely related to the unitary operator introduced by Kennedy and Tasaki.