engage has 19 distinct synonyms spread across 8 of its 10 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. Its one recorded opposite is disengage.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
engage or hire for work
“They hired two new secretaries in the department”
ask to represent; of legal counsel
“I'm retaining a lawyer”
Synonyms, antonyms and the broader and narrower relations for engage come from Princeton University’s WordNet. Full sources and licences.