Why are clothes “hung” but men “hanged”?

This old reference in the Dictionary of the English Language (thanks, Google Books!) supports the usage you outline as correct.

Grammar Girl writes that this is because there are 2 different Old English words for the two meanings:

[…] in Old English there were two different words for hang (hon and hangen), and the entanglement of these words (plus an Old Norse word hengjan) is responsible for there being two past-tense forms of the word hang today (1).

[Quote reference: Burchfield, R. W., ed. The New Fowler's Modern English Usage. Third edition. New York: Oxford, 1996, p. 349.]

According to the OED, the verb hang came into English from Old Norse hengja with weak inflection (so, taking regular past forms). Eventually, by analogy with other ablaut forms like sing/sang/sung, the verb hang changed into a few different forms (depending on the region of England), e.g. hing/hang, hang/hong, etc. Ultimately, the hing/hang form added hung to complete the sing/sang/sung analogy.

Now here's the important bit, directly from the OED:

The weak inflexion hanged however continued in use (being the only one used in Bible versions from Coverdale to 1611, though Tindale had also houng); but was gradually superseded by hung in the general sense, trans. and intr., leaving hanged only in the special trans. sense (3) ‘put to death by hanging’, owing prob. to the retention of this archaic form by judges in pronouncing capital sentences. The distinction is found already in Shakespeare, and is established in the objurgatory expressions ‘You be hanged!’ ‘I'll be hanged if I do’, and the like.

(At some point, obviously hing fell out of use in Standard English in the present tense form.)