c. 1300, "coupling device that allows independent rotation," from frequentative form of stem of Old English verb swifan "to move in a course, revolve, sweep" (a class I strong verb), from Proto-Germanic *swif- (source also of Old Frisian swiva "to be uncertain," Old Norse svifa "to rove, ramble, drift"), from PIE root *swei- (2) "to turn, bend, move in a sweeping manner."
Related Middle English swive was the principal slang verb for "to have sexual intercourse with," a sense that developed c. 1300. This probably explains why, though the root is verbal, the verb swivel is not attested in Modern English until 1794. Compare Middle English phrase smal-swivinge men "men who copulate infrequently."
1794 (transitive), from swivel (n.). Intransitive use from 1846. Related: Swiveled; swiveling; swivelled; swivelling.