in reference to opposite aspects of a person's character is a reference to Robert Louis Stevenson's story, "The Strange Case of Dr. Jekyll and Mr. Hyde," published in 1886. Jekyll, the surname of the respectful and benevolent man, is of Breton origin and was originally a personal name. Hyde in reference to the dark, opposite side of one's personality is from 1887.
"Though so profound a double-dealer, I was in no sense a hypocrite. Both sides of me were in dead earnest; I was no more myself when I laid aside restraint and plunged in shame, than when I labored, in the eye of day, at the furtherance of knowledge or the relief of sorrow and suffering." [Robert Louis Stevenson, "The Strange Case of Dr. Jekyll and Mr. Hyde," 1886]
The women's order, though sometimes persecuted, generally preserved its good reputation, but it quickly drew imposters who did not; nonetheless it eventually was condemned as heretical. A male order, called Beghards founded communities by the 1220s in imitation of them, but they soon degenerated (compare Old French beguin "(male) Beguin," also "hypocrite") and wandered begging in the guise of religion; they likely were the source of the words beg and beggar, though there is disagreement over whether Beghard produced Middle Dutch beggaert "mendicant" or was produced by it. The male order was condemned by the Church early 14c. and vanished by mid-16c.
Cole Porter's "Begin the Beguine" (1935) refers to a kind of popular dance of West Indian origin, from French colloquial béguin "an infatuation, boyfriend, girlfriend," earlier "child's bonnet," and before that "nun's headdress" (14c.), from Middle Dutch beggaert, ultimately the same word as the above. Compare English biggin "child's cap" (1520s), from the French word.