I've been doing some reading on ribbon categories, and something that's caught my attention lately is the ribbon category with half twist, see for example here. From my understanding, the objects in the category can be imagined as ribbons. Functors connect ribbons, and ribbons can fuse as well. The half twists and full twists correspond to twists in the ribbon, and obey some coherence relations with regard to braiding.
Here's my fundamental question(s): What does it mean for a ribbon to have a half-twist? When we feed in the half-twisted ribbon into a functor, does it give the same result as the non-twisted ribbon? If the ribbon is denoted by a vector space $A$, is the half-twist simply an isomorphism from $A \to A$? Or is the flip side of the ribbon a completely different vector space, $A'$? I know that the trace of a ribbon with half-twist, a Mobius strip, is not allowed in the formalism. Is $A' \otimes B \otimes C$ related to $A \otimes B \otimes C$? What about the F symbols? Are they the same for $A$ and $A'$?