This article gives a reformulation of the simplex method for quadratic programming having the advantage of generating tableaux with certain symmetry properties. It is proved that this method gives the same sequence of iterations as formulation of the simplex method for quadratic programming given earlier by Dantzing and the authors, which may be called the asymmetric formulation. A proof of convergence to the optimal solution is given, which is much simpler than the corresponding proof for the asymmetric formulation. A second symmetric formulation, which is equivalent to the two other formulations, is indicated. For the usual method for quadratic programming as well as for parametric quadratic programming, similar formulations exist. Wolfe's long form of his method for quadratic programming turns out to be the asymmetric variant for parametric quadratic programming.