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A cyclic triple, \( (a, b, c) \), is defined to be the set \( \{(a, b), (b, c), (c, a)\} \) of ordered pairs. A Mendelsohn triple system of order \( v \), or MTS\( (v) \), is a pair \( (M, \beta) \), where \( M \) is a set of \( v \) points and \( \beta \) is a collection of cyclic triples, each containing pairwise distinct points of \( M \) such that every ordered pair of distinct points of \( M \) exists in exactly one cyclic triple of \( \beta \). An antiautomorphism of a Mendelsohn triple system \( (M, \beta) \) is a permutation of \( M \) which maps \( \beta \) to \( \beta^{-1} \), where \( \beta^{-1} = \{(c, b, a) \mid (a, b, c) \in \beta\} \). Necessary conditions for the existence of an MTS\( (v) \) admitting an antiautomorphism consisting of two cycles of lengths \( M \) and \( N \), where \( 1 < M \leq N \), have been shown, and for the cases of \( N = M \) and \( N = 2M \), sufficiency has been shown. We show sufficiency for the cases in which \( M = 13 \) and \( N = 78, 390, \) and \( 702 \).