Quantum resource theories seek to quantify sources of nonclassicality that bestow quantum technologies their operational advantage. Chief among these are studies of quantum correlations and quantum coherence. The former isolates nonclassicality in the correlations between systems, and the latter captures nonclassicality of quantum superpositions within a single physical system. Here, we present a scheme that cyclically interconverts between these resources without loss. The first stage converts coherence present in an input system into correlations with an ancilla. The second stage harnesses these correlations to restore coherence on the input system by measurement of the ancilla. We experimentally demonstrate this interconversion process using linear optics. Our experiment highlights the connection between nonclassicality of correlations and nonclassicality within local quantum systems and provides potential flexibilities in exploiting one resource to perform tasks normally associated with the other.