Agile maneuvering of the quadrotor cable-suspended system is significantly hindered by its non-smooth hybrid dynamics. While model-free Reinforcement Learning (RL) circumvents explicit differentiation of complex models, achieving attitude-constrained or inverted flight remains an open challenge due to the extreme reward sparsity under strict orientation requirements. This paper presents ASTER, a robust RL framework that achieves, to our knowledge, the first successful autonomous inverted flight for the cable-suspended system. We propose hybrid-dynamics-informed state seeding (HDSS), an initialization strategy that back-propagates target configurations through physics-consistent kinematic inversions across both taut and slack cable phases. HDSS enables the policy to discover aggressive maneuvers that are unreachable via standard exploration. Extensive simulations and real-world experiments demonstrate remarkable agility, precise attitude alignment, and robust zero-shot sim-to-real transfer across complex trajectories.