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It is well known that “coordinating conjunctions” form a subconstituent with the phrase that follows them ([conj X]). We show that such a “conjunct phrase” can have different grammatical functions: it can be coordinated in a coordinated phrase, it can be adjoined to several categories, it can also appear as an independant sentence. We focus on French incidental and emphatic conjunct phrases: Jean est parti, et rapidement. (‘Jean has left, and rapidly’) Jean est en avance, ou je me trompe. (‘Jean is early, or I’m wrong’)Contrary to their properties as an element of a coordination, these conjunct phrases are mobile, do not trigger plural agreement and do not obey the constraint on parallel extraction. We analyze them as syntactic adjuncts, and extend the same analysis to phrases introduced by car in French, and to asymmetric verbal “coordinations” in Korean.We formalize our analysis with the phrase structure grammar HPSG, and propose to analyze conjunctions as “weak heads”, sharing most of their syntactic features with the following phrase.