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A graph has the neighbour-closed-co-neighbour, or ncc property, if for each of its vertices \(x\), the subgraph induced by the neighbour set of \(x\) is isomorphic to the subgraph induced by the closed non-neighbour set of \(x\). Graphs with the ncc property were characterized in [1] by the existence of a locally \(C_4\) perfect matching \(M\): every two edges of \(M\) induce a subgraph isomorphic to \(C_4\). In the present article, we investigate variants of locally \(C_4\) perfect matchings. We consider the cases where pairs of distinct edges of the matching induce isomorphism types including \(P_4\), the paw, or the diamond. We give several characterizations of graphs with such matchings. In addition, we supply characterizations of graphs with matchings whose edges satisfy a prescribed parity condition.