The existence of reverse Steiner triple systems (i.e. Steiner triple systems with a given involutory automorphism of special type) is investigated. It is shown that such a system exists for all orders n if n 1 or 3 or 9 (mod 24) except possibly for n = 25. A system with this property exists also for n = 19 and possibly for every n 19 (mod 24). On the other hand, it is demonstrated that such systems do not exist for the other values of n.