Is there a fusion category with an object which does not commute with its dual?

Does there exist a fusion category with an object $X$ such that $XX^*\ncong X^*X$ (where the isomorphism need not be natural in any way)?

Feel free to add adjectives such as pivotal, spherical, unitary, etc.

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If you allow "infinite depth" things then the universal example (oriented Temperley-Lieb) is a counterexample. –  Noah Snyder May 26 '11 at 19:12