Many dexterous manipulation tasks require the object to remain securely held throughout the interaction. From the perspective of hand-object relational motion, such manipulation comprises four canonical skills: grasping, relocation, in-hand rotation, and in-hand translation. Human hands flexibly compose these skills to accomplish complex tasks. Existing approaches, however, model these skills separately with skill-specific action constraints, objectives, or even dedicated hand morphologies, which breaks the compatibility and continuity required for long-horizon composition. In this work, we present a unified framework that models all four skills in a single formulation instantiated with desired relational targets. The formulation preserves cross-skill compatibility and enables distillation of a single policy without skill identifiers, which achieves strong per-skill performance and generalizes to unseen objects. The compatibility further supports the policy to perform long-horizon skill chaining and execute relational targets unseen during training. The framework also applies across different hand morphologies. Together, these results suggest that distinct dexterous manipulation skills can be modeled as instantiations of a shared relational task formulation. Project page: https://zdchan.github.io/UniCross/