This paper proposes a versatile mapping approach that has three objectives: i) it can map from one technology-independent graph representation to another; ii) it can map to a cell library; iii) it supports logic rewriting. The method is cut-based, mitigates logic-sharing issues of previous graph mapping approaches, and exploits structural hashing. The mapper is the first one of its kind to support remapping among various graph representations, thus enabling specialized mapping to emerging technologies (such as AQFP) and for security applications (such as XAG-based design). We show that mapping to MIGs improves area by 10% as compared to the state of the art, and that technology mapping is 18% faster than ABC with slightly better results.