Due to update the Lagrangian multiplier twice at each iteration, the symmetric alternating direction method of multipliers (S-ADMM) often performs better than other ADMM-type methods. In practice, some proximal terms with positive definite proximal matrices are often added to its subproblems, and it is commonly known that large proximal parameter of the proximal term often results in ``too-small-step-size" phenomenon. In this paper, we generalize the proximal matrix from positive definite to positive-indefinite, and give a new S-ADMM with positive-indefinite proximal regularization (IPS-ADMM) for the two-block separable convex programming with linear constraints. Without any additional assumptions, we prove the global convergence of the IPS-ADMM and analyze its convergence rate under the ergodic sense by the iteration complexity. Finally, numerical results reveal that the IPS-ADMM is more efficient than some ADMM-type methods with positive definite proximal regularization.