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“commute” in Latin

The English word commute itself goes back to Latin mūtō, through commūtō.

Riddle & Arnold A Copious and Critical English–Latin Lexicon, 1864

commute
mutare, for anything (cum) aliqua re (to change): permutare, for anything, aliqua re (exchange, especially with regard to bills or barter): commutare cum aliqua re (to change, i. e., to put one thing in the place of the other) to commute things, res inter se mutare or permutare.

Quick answer

What is the Latin word for commute?
The usual Latin words for “commute” are mūtō, permūtō. The right choice depends on the sense; see the distinctions below.

Entry from J. E. Riddle & T. K. Arnold, A Copious and Critical English–Latin Lexicon (rev. C. Anthon, New York: Harper, 1864), digitised by Fergus J. P. Walsh, CC BY-SA 4.0; Latin markup adapted by latindictionary.co.