Computer systems are prone to crashes and power outages, particularly in intermittent computing scenarios where embedded systems rely on unreliable energy-harvesting sources. Hybrid caches, which integrate volatile and non-volatile memory (NVM) sections, offer opportunities for efficient operating system (OS) services to ensure data persistence over power interruptions. However, the lack of OS awareness of hybrid cache structures can result in inefficient backups, data loss and inconsistencies, as well as underutilization of the potential of hybrid caches. This paper introduces the Cache Hybridization-aware Operating System (CHaOS), that is based on a co-design of the OS and the underlying hardware to systematically persist volatile states and data in case of power outages. Our evaluation demonstrates that CHaOS not only proves robust against power outages, but also can significantly reduce the backup latency. In our test of using CHaOS to execute an image processing application on a hybrid-cache system, already with 75% of caches being non-volatile, the backup latency drops to only 8.2% compared to a completely volatile cache. At the same time, the energy demand could be reduced due to lower static power consumption of NVM technology. Our investigations validate the feasibility of CHaOS and establish a strong foundation for future research.