As Scott points out, Peter Tingley and I wrote a paper about this question. For the sake of concreteness (and because it included our main example) we only deal with the case of Hopf algebras whose representation theory has a ribbon half-twist, but the whole theory carries over to general monoidal categories. I'll sketch this generalization below, but you should probably read my paper with Peter first which is more accessible. The main result we use are formulas for the braiding given (independently) by Kirillov-Reshetikhin and Levendorskii-Soibleman which can be interpreted as given a formula for the half-twist. Certainly Reshetikhin (and presumably some of the other authors) were aware that these formulas could be interpreted in terms of half-twists, but it didn't explicitly appear until Peter and my paper.

One important warning that applies to everything though, in this theory the front and the back of the ribbon correspond to a priori different objects, so you still can't talk about Mobius bands.

Recall that a monoidal functor (these are often called "weak monoidal functors" in the quantum groups literature to distinguish them from strict monoidal functors, while they're called "strong monoidal functors" in the category theory literature to distinguish them from "lax monoidal functors) is a pair a functor F: C->D together with a binatural transformation $F(X\otimes Y) \rightarrow F(X) \otimes F(Y)$ which plays well with the associator.

Let's define a commutor to be a monoidal functor from C to C' (which will denote C with the opposite tensor product) whose underlying functor is the identity. The natural transformation is thus a map $X \otimes Y \rightarrow Y \otimes X$. The consistency condition says that there's a well-defined map $X \otimes Y \otimes Z \rightarrow Z \otimes Y \otimes X$. This definition of a commutor is the common generalization of braidings (which additionally satisfy the Yang-Baxter equation) and cactus commutors (which additionally square to 1). It's a natural condition that is satisfied by all known interesting "commutivity constraints."

There is a common way to produce commutors which comes up in a paper of Kamnitzer and Henriques (which is a very beautiful paper) which is closely related to half-twists.

First let's define a "dark side functor" to be any monoidal functor from C to C'. The name comes from the fact that this functor is what lets us talk about the "dark side" of the ribbon. If F is a dark side functor, then a half-twist for F is a natural transformation between F and the identity functor from C to C'.

Certainly you can use a half-twist for a dark side functor to produce a commutor, just compose the natural transformation with the dark side functor to get a commutor! If you work through what this tautological explanation means, you'll see that you get the commutor by first applying the half-twist to each object seperately, and then the inverse of the half-twist to the tensor product (or maybe visa-versa).

A ribbon half-twist is just a half-twist for a dark side functor whose resulting commutor is a braiding.

As Scott points out, Peter Tingley and I wrote a paper about this question. For the sake of concreteness (and because it included our main example) we only deal with the case of Hopf algebras whose representation theory has a ribbon half-twist, but the whole theory carries over to general monoidal categories. I'll sketch this generalization below, but you should probably read my paper with Peter first which is more accessible. The main result we use are formulas for the braiding given (independently) by Kirillov-Reshetikhin and Levendorskii-Soibleman which can be interpreted as given a formula for the half-twist. Certainly Reshetikhin (and presumably some of the other authors) were aware that these formulas could be interpreted in terms of half-twists, but it didn't explicitly appear until Peter and my paper.

One important warning that applies to everything though, in this theory the front and the back of the ribbon correspond to a priori different objects, so you still can't talk about Mobius bands.

Recall that a monoidal functor (these are often called "weak monoidal functors" in the quantum groups literature to distinguish them from strict monoidal functors, while they're called "strong monoidal functors" in the category theory literature to distinguish them from "lax monoidal functors) is a pair a functor F: C->D together with a binatural isomorphism $F(X\otimes Y) \rightarrow F(X) \otimes F(Y)$ which plays well with the associator.

Let's define a commutor to be a monoidal functor from C to C' (which will denote C with the opposite tensor product) whose underlying functor is the identity. The natural transformation is thus a map $X \otimes Y \rightarrow Y \otimes X$. The consistency condition says that there's a well-defined map $X \otimes Y \otimes Z \rightarrow Z \otimes Y \otimes X$. This definition of a commutor is the common generalization of braidings (which additionally satisfy the Yang-Baxter equation) and cactus commutors (which additionally square to 1). It's a natural condition that is satisfied by all known interesting "commutivity constraints."

There is a common way to produce commutors which comes up in a paper of Kamnitzer and Henriques (which is a very beautiful paper) which is closely related to half-twists.

First let's define a "dark side functor" to be any monoidal functor from C to C'. The name comes from the fact that this functor is what lets us talk about the "dark side" of the ribbon. If F is a dark side functor, then a half-twist for F is a natural transformation between F and the identity functor from C to C'.

Certainly you can use a half-twist for a dark side functor to produce a commutor, just compose the natural transformation with the dark side functor to get a commutor! If you work through what this tautological explanation means, you'll see that you get the commutor by first applying the half-twist to each object seperately, and then the inverse of the half-twist to the tensor product (or maybe visa-versa).

A ribbon half-twist is just a half-twist for a dark side functor whose resulting commutor is a braiding.