In-hand manipulation is an integral component of human dexterity. Our hands rely on tactile feedback for stable and reactive motions to ensure objects do not slip away unintentionally during manipulation. For a robot hand, this level of dexterity requires extracting and utilizing rich contact information for precise motor control. In this paper, we present AnyRotate, a system for gravity-invariant multi-axis in-hand object rotation using dense featured sim-to-real touch. We construct a continuous contact feature representation to provide tactile feedback for training a policy in simulation and introduce an approach to perform zero-shot policy transfer by training an observation model to bridge the sim-to-real gap. Our experiments highlight the benefit of detailed contact information when handling objects with varying properties. In the real world, we demonstrate successful sim-to-real transfer of the dense tactile policy, generalizing to a diverse range of objects for various rotation axes and hand directions and outperforming other forms of low-dimensional touch. Interestingly, despite not having explicit slip detection, rich multi-fingered tactile sensing can implicitly detect object movement within grasp and provide a reactive behavior that improves the robustness of the policy, highlighting the importance of information-rich tactile sensing for in-hand manipulation.