<p>Symmetry is a recurring feature in algorithms for monotone operator theory and convex optimization, particularly in problems involving the sum of two operators, as exemplified by the Peaceman–Rachford splitting scheme. However, in more general settings–such as composite optimization problems with three convex functions or structured convex-concave saddle-point formulations–existing algorithms often exhibit inherent asymmetry. Notably, while both the Condat–Vũ algorithm and the asymmetric forward-backward-adjoint (AFBA) method are efficient and widely used, they apply extrapolation only to either the primal or dual updates, leading to unbalanced iterative structures. In this work, we introduce a symmetric primal-dual algorithm (SPDA) that preserves symmetry in the iterative scheme by applying extrapolation to both primal and dual iterates. The algorithm encompasses the Condat–Vũ and AFBA methods as special cases and permits more flexible step-size choices. We establish global convergence under standard assumptions and derive both ergodic and non-ergodic convergence rates. The results demonstrate that symmetry can be preserved in first-order methods for optimizing the sum of three convex functions without compromising convergence guarantees or practical simplicity.</p>