After my comment on how my mantion is significantly shorter in the back than on the sides, I started looking for images of people wearing semicircular cloaks in Middle Byzantine and Byzantine-influenced art.
I didn't find as many as I'd hoped. I did find several images where figures wearing similar-looking cloaks appear to have the sides hitched up over their hands, as does the Virgin Mary in this 12th-13th century Russian icon. The left-hand figure in this Crucifixion scene of similar date (which probably also is Mary) is doing the same thing. But wait; this Mary appears to have a rectangular garment wrapped about her like a Roman Imperial era palla, which wasn't a living fashion by the Middle Byzantine era. It's hard to tell what the saint on the right here is wearing--his cloak is so dark it is difficult to see where the folds are and how it is being worn.
This 10th century manuscript of the Gospel of Luke looks as though the figures are wearing more contemporary clothing, but again the one figure who might be wearing a semicircular cloak (lower right) cannot be clearly discerned. And contemporary art from north or east of Byzantium is of no help; all of the semicircular cloaks shown are too long and voluminious for any difference in length on the sides to be discernible. This page on medieval material culture has a collection of secular images, but aside from the fact that they are not from the Byzantine Empire most of them are too long to ascertain whether there is any difference in hem length at the sides. Most of them also date from the 14th century or later--significantly after the Middle Byzantine period.
To make things more confusing, this illumination from the Fecamp Psalter shows a semicircular cloak that is shorter on the sides. Some of the Russian icon images look shorter on the sides, but that appears to be because the wearer has them wound around his/her hands or is holding them in some way. The Fecamp image shows the cloak being caught up by the arms a bit, but even so, the cloak's sides look much too short for that factor to account for all of the difference in length.
Interestingly enough, there are a few actual surviving 12th century ceremonial cloaks (from Germany, so far as I know). Some of them are almost pure half-circles, without even the broad, slightly curved neck notch I eventually made. Here's another one; the edge that goes against the neck is perfectly straight, though the sides are chopped off a bit. But wait! The ceremonial mantle of the Holy Roman Emperor has both a neck notch *and* loops for a fastener, and the loops are at the corners of the notch, where I put mine! (And the lion applique on it is stunning, too.)
So although I can't say at this point that I've proven my design to be "period" for the Middle Byzantine era, the material I've found suggests that there may be a larger spectrum of designs for period semicircular cloaks than I'd thought. When I get a chance, I need to look more systematically for images of cloaked figures from the period.