A \textit{hypermap} is a representation of a hypergraph in a closed connected surface; a generalization of the concept of a topological map. The aim of this paper is to develop a measure of how far a hypermap is from being self-dual. This is done by understanding a hypermap algebraically as a transitive permutation representation on the set of its hyperflags, and defining the \textit{duality group} of a hypermap as certain conjugation of this representation by an outer automorphism. In case of an oriented regular hypermap (a hypermap with the highest possible number of orientation preserving automorphisms), its hypermap subgroup is normal in its transitive representation, and the author shows that a number of different factors involving this hypermap subgroup are isomorphic. The \textit{duality index} of an oriented regular hypermap is then defined to be the order of these factor groups, and corresponds to the smallest self-dual hypermap covering the given hypermap and the largest self-dual hypermap covered by it (if the duality index of a hypermap is \(1\), the three hypermaps coincide and the hypermap is self-dual). The main contributions of this article include showing the existence of a self-dual oriented regular hypermap for every order \(k\) (based on cyclic groups), the existence of an oriented regular hypermap for every duality index \(d\), and the existence of an oriented regular hypermap for every duality co-index \(k\) (the ratio of the order of the group of the hypermap and its duality index). Further results concerning oriented direct products of hypermaps and hypermaps based on generalized quaternion groups are also included. The duality index considered in this article is developed along the lines of the chirality index developed previously by other authors (the chirality index measures how far a hypermap is from being chiral) and some of the arguments are simply a repetition of those for the chirality index. It also needs to be noted that hypermaps allow for several different dualities, and thus the results obtained in this paper all refer to the specific duality considered in the paper.