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Many representation learning problems involve directed relations, such as lexical entailment, sentence entailment, ontology hierarchy, and citation links. Standard Euclidean, cosine, and Mahalanobis heads are symmetric, while generic neural scorers can model directionality but provide limited geometric structure. This paper proposes a role-aware neural convex divergence head for asymmetric representation learning. The head applies source- and target-role projections before evaluating an input-convex neural Bregman divergence, yielding a nonnegative structured score in the role-projected space. We characterize its projected-space identity, source-role convexity, directional-gap decomposition, and Hessian-based local curvature. Experiments on lexical, sentence, ontology, and directed graph benchmarks compare symmetric distances, unstructured asymmetric scorers, order/hyperbolic baselines, plain ICNN-Bregman heads, and the proposed role-aware variant. Across ten random seeds on the main semantic and ontology benchmarks, role-aware projections consistently improve directional accuracy over plain ICNN-Bregman heads while preserving zero observed negative divergence rate. The results also identify a boundary case: on large fixed-feature citation prediction, specialized symmetric or hyperbolic baselines remain stronger in ranking accuracy. Overall, the proposed head is best understood as a structured and interpretable plug-in distance module for tasks where directional relations matter.