P.M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a; b is an element of R. In this paper, we study an extension of a reversible ring with its endomorphism. An endomorphism alpha of a ring R is called strong right (resp., left) reversible if whenever a alpha(b) = 0 (resp., alpha(a)b = 0) for a; b is an element of R, ba = 0. A ring R is called strong right (resp., left) alpha-reversible if there exists a strong right (resp., left) reversible endomorphism alpha of R, and the ring R is called strong alpha-reversible if R is both strong left and right alpha-reversible. We investigate characterizations of strong alpha-reversible rings and their related properties including extensions. In particular, we show that every semiprime and strong alpha-reversible ring is alpha-rigid and that for an alpha-skew Armendariz ring R, the ring R is reversible and strong alpha-reversible if and only if the skew polynomial ring R[x; alpha] of R is reversible.