Wrong end of the stick
From the 16th century onwards, this expression, the wrong or short end of the stick meant getting the worst end of a deal or bargain and it was not until the 19th century that it took on its alternative meaning of being mistaken or misunderstanding something. Quite why a stick should have a right end as opposed to a wrong end, no one knows for sure. One theory is that it derives from the 16th century game of billiards where clearly using the wrong end of the stick would be a mistake. Another is that it refers to walking sticks, which also have a right end and a wrong end. Unfortunately, there is no definitive answer and the only thing that is certain is its 16th century origin.