Regarding the question of how idempotent elements affect the reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce right (resp., left) e-reversible rings. We show that this concept is not left-right symmetric. Basic properties of right e-reversibility in a ring are provided. Among others, it is proved that if R is a semiprime ring, then R is right e-reversible if and only if it is right e-reduced if and only if it is e-symmetric if and only if it is right e-semicommutative. Also, for a right e-reversible ring R, R is a prime ring if and only if it is a domain. It is shown that the class of right e-reversible rings lies strictly between that of e-symmetric rings and right e-semicommutative rings. Some extensions of rings are studied in terms of e-reversibility.