We propose an iterative method named USYMLQR for the solution of symmetric saddle-point systems that exploits the orthogonal tridiagonalization method of Saunders, Simon, and Yip (1988). By contrast with methods based on the Golub and Kahan (1965) bidiagonalization process, our method takes advantage of two initial vectors and splits the system into the sum of a least-squares and a least-norm problem. In our numerical experiments, USYMLQR typically requires fewer operator-vector products than MINRES, yet performs a comparable amount of work per iteration and has comparable storage requirements.