The most common and most likely answer to this crossword clue is the 11 letter word SIDE-SADDLE
I've seen this crossword clue in Financial Times
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More clues leading to the result SIDE-SADDLE
Buddy explains!
The straight part
We know that "when not sitting up straight" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case when not sitting up straight = side-saddle.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SIDE-SADDLE mean?
SIDE-SADDLE noun- A saddle, usually for a woman, in which the rider sits with both legs on the same side of the horse.
- On, or as if on, such a saddle.