"Couth", meaning sophisticated or refined, is a back-formation from "uncouth"; it's sometimes used in a mocking or ironic manner. The reasoning runs that if "uncouth", meaning crude or unrefined, is a word, then surely there needs be a word "couth" to be negated by the prefix "un".
Websters 1913 gives us the original meanings of both "couth" and "uncouth". It is "uncouth" that has evolved in its meaning, from unfamiliar to unfriendly to awkward/crude/boorish/unrefined. "Couth" was left behind to wither in obsolescence until given new life and new meaning by the truncation of "uncouth".
Is this the only fate of "couth", to be lost then resurrected so far from its original meaning? Happily no, for "couth" lives on through an etymological descendant in Scottish dialect as "couthie" (also spelled "couthy"), meaning friendly or genial.
Everything I know about etymology, I learned from OED, compact edition.