I fixed one thing: the axioms for a tortile category states that the twist is self-dual, not the braiding (that is already a consequence of the axioms of braided autonomous categories).

Thanks. Where I come from, a lot of people call the braiding the "twist map", and it didn't occur to me that this is already necessarily self-dual. By "twist", you mean the same structure in a balanced monoidal category that is also called the "balance", right? --Toby