"Tit for tat"—Where does this come from?

Tit and tat are used here to mean striking a light blow, so the phrase has exactly the same meaning as blow for blow. They were used as both nouns and verbs, as a sixteenth-century rhyme shows:

Come tit me, come tat me,
come throw a kiss at me.

(Source; An earlier variation has halter instead of kiss.)

I always thought it was a cute mispronunciation of this for that, but apparently it's actually a corruption of tip for tap (Etymonline; concise etymological dictionary), with the same—and, I should think, slightly more obvious—meaning.

An exhaustive derivation from http://www.businessballs.com/clichesorigins.htm:

tit for tat - retribution or retaliation, an exchange insults or attacks - 'tit for tat' evolved from 'tip for tap', a middle English expression for blow for blow, which also meant a trade of verbal insults. Tit is an old English word for tug or jerk. Tat evolved from tap partly because of the alliteration with tit, but also from the verbal argument aspect, which drew on the influence of the Middle English 'tatelen' meaning prattle, (Dutch tatelen meant stammer) which also gave rise to tittle-tattle. Tip and tap are both very old words for hit. (eg 'tip and run' still describes a bat and ball game when the player hits the ball and runs, as in cricket). Tit for tat was certainly in use in the mid-late 16th century. Tip for Tap was before this. As with lots of these old expressions, their use has been strengthened by similar sounding foreign equivalents, especially from N.Europe, in this case 'dit vor dat' in Dutch, and 'tant pour tant' in French. Skeat's 1882 dictionary of etymology references 'tit for tat' in 'Bullinger's Works' . Brewer in 1870 suggests for 'tit for tat' the reference 'Heywood', which must be John Heywood, English playwright 1497-1580 (not to be confused with another English playwright Thomas Heywood 1574-1641). According to James Rogers dictionary of quotes and cliches, John Heywood used the 'tit for tat' expression in 'The Spider and the Flie' 1556.