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 \(\lambda \geq 1\).