v. to part or divide by cutting asunder. (OED)
v. to stick fast or to adhere (OED)
My friend, and a very talented musician, Sandra R.B. suggested this word as a possible auto-antonym, and it was one of two that I am readily familiar with. I was at a wedding of a college roommate of mine in Vermont when I was first stumped by this. Her wedding vows consisted of something akin to “to cleave on to you for the rest of my life.” The phrase was quite jarring, considering that my only understanding of the word was the first definition above. How could a word mean to rip asunder and to adhere together yet still have any meaning? Never mind the fact that, as a noun, it also means an Irish basket.
This is a rare case of a true auto-antonym, as opposed to one that was formed from misuse. The word is actually two different words, stemming from two different etymologies, that happen to be spelled and pronounced the same way and have meanings that contradict each other.
The two words are not necessarily identical in their other forms. In both cases you can say that you cleaved something or cleaved to something, but other forms of the past tense exist that are more specific. So the retired lumberjack clove wood and cleft wood, while the divorced couple clave each other. Wood is cloven while a contract is cleaved to. In the end, these two words at some point just became too close to each other, and could only be cloven when put into different tenses.
For the record, my Vermont roommate still cleaves to her husband. They are building a beautiful inn, called the Snapdragon in Windsor, VT, and you can read about its progress here. The also have a delicious looking eatery called the Windsor Station Pub. If you are ever in Windsor, cleave to those locations.
P.S. Cleave is also a noun form of the word cleft. According to the OED, the first published use of the noun cleavage was in Time magazine’s August 1946 issue. It was in reference to a trade term about the cut of an actress’s dress.