This study is conceived as a contribution to a typology of attributive constructions, focusing on constructions exhibiting splits of head properties. The term "dependency reversal in noun-attributive constructions" (DNRA) is used to refer to possessive-like attributive constructions (of the type (that) idiot of a doctor), with the attribute surfacing as the formal head and the semantic head surfacing as the formal possessor. The body of the study presents a discussion of DNRA constructions as attested in six individual languages: Even (resp. Other Tungusic languages), Aleut, Hausa, Gude, Chinook and Latin. The variation of the DNRA patterns, in particular along the parameter of the attribute's upgrading/recategorization, is further considered. Following the lines of structure-based typologies, an upward taxonomy of DNRA structures is presented to include other cases of constructions involving the attribute's upgrading and the head (-to-possessor) demotion. In search of DNRA related patterns the discussion is extended to internal relative clauses and constructions with "dominant attributes". Finally factors favouring the rise of DNRA structures are tentatively considered: apart from diachronic factors, underdifferentiation of lexical categories as well as pragmatic salience of the attribute are shown to contribute to DNRA processes.