Quinion writes “It became inverted around the end of the eighteenth century, it seems as the result of a series of mistakes by authors who didn’t stop to think about the conventional phrase they were writing.”
In the earlier discussion to which Dr. Techie kindly provided a link, there is a query (unanswered) about a supposed British expression “ass over teacups”. I never heard that, but do recall from my days in Britain “arse over tip”, as in “he took a tumble, arse over tip”. I wonder why “tip”? Is it perhaps an archaic usage for “head”? Or possibly for “hat”? Or simply a synonym for “top” (as in “tip of the iceberg")?
… Ass over teakettle is one of many variants of an expression meaning ‘head over heels; topsy-turvy; in confusion’. The usual British version is ass over tip (or tit), which occurs in James Joyce’s Ulysses, among other works. This form also occurs in America. For instance, in The Grapes of Wrath Steinbeck has a character say “You jus’ scrabblin’ ass over tit, fear somebody gonna pin some blame on you.” The earliest known example of the phrase is in an 1899 book about Virginia folk exressions, which defines “ass over head” as “Head over heels; topsy-turvy.”