Do “retention” and “engagement” mean the same thing when law departments hire law firms?

The pedant in me stubbornly insists that English has no exact synonyms, so off to the sophistry of hair splitting I must go. If “retention” carries with it a notion of a retainer – an amount paid each month against which fees are charged, then perhaps an “engagement” is simpler: “We hired you to represent us.” Then, oh readers who enjoy the rabbit hole into verbal Wonderland, what sorts out “engage” from “hire”? The former sounds snootier; hire sounds like day workers.