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1590s, "to turn a ship on its side" (with the keel exposed), from French cariner, literally "to expose a ship's keel," from Middle French carene "keel" (16c.), from Italian (Genoese dialect) carena, from Latin carina "keel of a ship," originally "nutshell," possibly from PIE root *kar- "hard" (see hard (adj.)).
Intransitive sense of "to lean, to tilt" is from 1763, specifically of ships; in general use by 1883. In sense "to rush headlong," confused with career (v.) since at least 1923. [To career is to move rapidly; to careen is to lurch from side to side (often while moving rapidly).] Earlier figurative uses of careen were "to be laid up; to rest." Related: Careened; careening.