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The company does not have a legal right to require you to produce your books for inspection if it is not in your contract with them. It sounds like the invoices that you submitted detailing the hours you worked and showing what you did should be enough.
However, if the company were to sue you alleging overcharges, the company could ask for that information in "discovery." Discovery is done after a lawsuit is filed, and in writing. Responses are made under oath. You would have a right to object to discovery requests that were over broad, invaded your right to privacy, or the right to privacy of the other clients that you worked for.
I wish you the best with this situation.
My answer would be the same for an individual, sole proprietorship, "dba", LLC or LLP, or corporation.