\ys{This paper introduces} a symmetric version of the generalized alternating direction method of multipliers for two-block separable convex programming \ys{with linear equality constraints, which inherits the superiorities of the classical alternating direction method of multipliers (ADMM), and extends the feasible set of the relaxation factor $\alpha$ of the generalized ADMM to the infinite interval $[1,+\infty)$}. \ys{Under the conditions that the objective function is convex and the solution set is nonempty}, we \ys{establish} the \ys{convergence} results of the proposed method, including the global convergence, the worst-case $\mathcal{O}(1/k)$ convergence rate in both the ergodic and non-ergodic senses, where $k$ denotes the iteration counter. Numerical experiments to decode a sparse signal arising in compressed sensing are included to illustrate the efficiency of the new method.