This paper develops a unified information-theoretic framework for the characterization of continuous symmetric distributions based on extropy and cumulative extropy measures. The proposed approach is formulated through the structural properties of consecutive (n−r)-out-of- $n$:G and (n−r)-out-of- $n$:F systems, which naturally arise in reliability and lifetime analysis. We establish necessary and sufficient conditions under which the equality of differential extropy, cumulative residual extropy, and cumulative past extropy uniquely characterizes distributional symmetry for all $n\ge 2r$. These results contribute to statistical distribution theory by providing new system-based information characterizations of symmetry. Building on the theoretical developments, we introduce a new nonparametric test for symmetry constructed from differences between estimated extropy measures associated with consecutive systems. The proposed test is shown to be consistent, and extensive Monte Carlo simulations demonstrate that it achieves competitive and often superior power compared with several existing symmetry tests, particularly in detecting mild and near-symmetric departures. Applications to real datasets further confirm the practical effectiveness and interpretability of the proposed methodology. The proposed framework naturally admits extensions in multivariate settings and to other information measures.