Braid / Other Theory to Describe this version of the Dirac Belt Trick The video:
http://vimeo.com/62143283
supposedly shows a variation on the Dirac Belt trick.
This is most decidedly not even a variation on the standard belt trick. It is simply undoing a semi-infinite braid by straightening one link, thus leaving it with the other link wound once around the first, then looping the wind in the second link over the end of semi-infinite straightened link. Then, one exploits the fact that one can concatenate two semi-infinite braids f the above kind and impart the looping manoeuvre such that each is equivalent to a semi-infinite braid when the two looping manoeuvres reach the point where "loopover" happens.
The total braid here is the concatetation of a complete twist and its inverse, so is the identity braid. However, I don't think this is relevant because the above trick only works with one full twist, not a half twist (a concatenated half twists and its inverse would also be the identity!).
The individual belts do not undergo the homotopy imparted by the standard belt trick, so I don't think this trick, although superficially like the Dirac trick, is related to the modelling of the homotopy class of a path through $SO(3)$ by twists in a ribbon.
Is there a name for the kind of problem where we consider braids in $\mathbb{R}^3$ where there is are central fixed points in the braids, so that a braid on one side of this set of central points with its inverse on the other side is not considered the same as the "identity"? Is there a sub-theory of braids that considers this kind of problem?
 A: If we forget for the moment that these are ribbons, and instead consider them as strings, and if we forget one half of the picture, and replace the 'fixed points' by a single fixed sphere which is attached to both strings, this video is illustrating then that the sphere braid group on two strings has an element of order two (the half-full twist). This element is normally called $\sigma_1$ using the usual braid generator notation. This is how we would mathematically describe the usual 'Dirac belt trick'.
If you want to make the strings into ribbons again, then by considering the boundary of the ribbons as distinct strings, this video is also illustrating that the sphere braid group on four strings has an element of order 2 (the exchange of two pairs of strings). This element is normally called $\sigma_2\sigma_3\sigma_1\sigma_2$ using the usual braid generator notation.
A good reference for the braid groups on manifolds other than the disk (although this is also a good reference for the usual braid groups as well) is

J. S. Birman Braids, Links and Mapping Class Groups, Annals of Mathematical Studies 82, Princeton University Press, 1975.

I also like this book, which has a well-written account of the proof of torsion in the sphere braid groups in particular and is written with a slightly more modern approach

V. L. Hansen, Braids and Coverings, London Mathematical Society Student Texts 18, Cambridge University Press, 1989.

I should mention, this is interesting because among two-dimensional manifolds, only the sphere (and real projective plane if you want to consider non-orientable surfaces as well) admits braids with finite order - and in fact for any number of strings greater than one. All surfaces of higher genus (in both the orientable and non-orientable case) have torsion-free braid groups.
