If you didn't sign a contract either, then you can throw it back in their face just as much. Granted, it wouldn't be professional, but it sounds like it's already passed that point.
If it were me, I'd probably just tell them I'd refund their money but they don't have any rights to the design. (And I'd draft up an agreement and make them sign off on it.) I'd then just try to sell the design on a forum (removing their name and logos, of-course).
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