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Recently it was shown that the seminal Maddah-Ali and Niesen (MAN) coded caching scheme leaks the demand information of each user to the others. Many works have considered coded caching with demand privacy, while each non-trivial existing coded caching scheme with private demands was built on the fact that the cache information of each user is private to the others. However, most of these schemes leak the users' cache information. Consequently, in most realistic settings (e.g., video streaming), where the system is used over time with multiple sequential transmission rounds, these schemes leak demand privacy beyond the first round. This observation motivates our new formulation of coded caching with simultaneously private demands and caches. The main contribution of this paper is a new construction that generates private coded caching schemes by leveraging two-server private information retrieval (PIR) schemes. We show that if in the PIR scheme the demand is uniform over all files and the queries are independent, the resulting caching scheme is private on both the demands and on the caches. Interestingly, we propose a new construction of two-server PIR schemes in this class by leveraging coded caching schemes. By applying the seminal MAN coded caching scheme into our construction, the resulting two-server PIR scheme is proved to be order optimal. This is a second new structural result, somehow closing the loop in the relation between coded caching and PIR. Finally, to explore a broader tradeoff between cache privacy and transmission load, we relax the cache privacy constraint and introduce the definition of leakage on cache information. Then, again as a by-product of our new construction, we propose new schemes with perfect demand privacy and imperfect cache privacy that achieve an order-gain in load with respect to the scheme with perfect privacy on both demands and caches.