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For more on marking an answer as the "Best Answer", please visit our FAQ.The earliest recorded version was 'lean over backwards', which first appeared in print in 1925. In 1937, the 'bend over backwards' version emerged; both it and the earlier one were in American publications.
The earliest-recorded use of either in a British publication was in 'The Economist' magazine in November 1952. An article in that said:
"Broadmindedness may be carried beyond the optimum of impartiality to that point of unfairness to one's own case which Americans call 'leaning over backward'."
It is clear from that quote what the phrase means, where it came from and how very recent an import it is. It suggests you go out of your way to help someone, even to the extent of inconvenience to yourself.
There are some people who claim the phrase came from sailors unfurling sails on old sailing-ships, but that's obviously no more likely than that it came from rowing-boat crews or tug-o'-war teams, who also have to �bend over backwards' to achieve their aims. Given that the phrase appeared nowhere in print prior to about the mid-20th century, that is much too late, given Britain's long naval traditions and centuries of sea-tales before then, for any sailing connection.