We provide two alternative characterizations of hereditary history-preserving bisimilarity: a denotational one, on stable configuration structures, and an operational one, on a reversible process calculus. The characterizing equivalence is forward-reverse bisimilarity extended with a check for backward ready multiset equality. Unlike previous approaches, the focus is thus on counting identically labeled events rather than uniquely identifying them. We also investigate the relationships between event identifier logic, characterizing the former bisimilarity, and backward ready multiset logic, characterizing the latter bisimilarity.