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A directed triple system of order \(v\) and index \(\lambda\),
denoted \({DTS}_\lambda(v)\), is said to be reverse if it
admits an automorphism consisting of \(v/2\) transpositions when \(v\)
is even, or a fixed point and \((v-1)/2\) transpositions when \(v\)
is odd. We give necessary and sufficient conditions for the
existence of a reverse \({DTS}_\lambda(v)\) for all \(\lambda \geq 1\).