Holomorphic correspondences are multivalued functions from the Riemann sphere to itself. This thesis is concerned with a certain type of holomorphic correspondence known as a covering correspondence. In particular we are concerned with a one complexdimensional family of correspondences constructed by post-composing a covering correspondence with a conformal involution. Correspondences constructed in this manner have varied and intricate dynamics. We introduce and analyze two subfamilies of this parameter space. The first family consists of correspondences for which the limit set is a Cantor set, the second family consists of correspondences for which the limit set is connected and for which the action of the correspondence on the complement of this limit set exhibits certain group like behaviour.