I believe McGrath of Harlow was on the right track with his allusion to the phrase "get knotted". One of the meanings of the verb "to bend" is "to fasten"--for example, "The sailors bent a sail to the boom." "Sheet bend" and "carrick bend" are the names of knots. So the answer to the question "What does 'get bent' mean?" is, "It means exactly the same thing as 'get knotted'".
Next question: why "knotted"? Why not buttoned, zippered, velcroed, stapled, nailed, tied, glued, or fastened?
Perhaps someone with access to Partridge's Dictionary of Slang could look up "get bent" and "get knotted" and tell us which has the earlier citation.