Sequential optimality conditions have played a major role in proving strong global convergence properties of numerical algorithms for many classes of optimization problems. In particular, the way complementarity is dealt is fundamental to achieve a strong condition. Typically, one uses the inner product structure to measure complementarity, which gives a very general approach to a general conic optimization problem, even in the infinite dimensional case. In this paper we exploit the Jordan algebraic structure of symmetric cones in order to measure complementarity, which gives rise to a stronger sequential optimality condition. Our results improve some known results in the setting of semidefinite programming and second-order cone programming in a unified manner. In particular, we obtain global convergence results which are stronger than the ones known for augmented Lagrangian algorithms and interior point methods for general symmetric cones.