The most common and most likely answer to this crossword clue is the 8 letter word CANTORIS
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The straight part
We know that "Rigid" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case rigid = cantoris.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 CANTORIS mean?
CANTORIS adjective- Of the side of the chancel, apse, altar or choir on which the cantor's (later precentor's) stall is placed (the left hand side to a person facing the altar);