I have those. They are similar to roping reins. In fact, there is very little difference in these and the old roping reins I have, save they are pretty and match the rest of my tack. ;]
They are a single leather strap, unlike split reins with are two.
They have a snap on one end so that it have be unclipped and the horse can be led, but I don't see much use in unclipping it. It's just a convenience thing, and a clip could easily be added to the other side for easy removal.
Wow, that sounds like it would be a really good idea to 'permanently' hook one side to the bit, and then have the other rein just clipped on for easy leading I guess you could say, but I don't know when you would ever unclip the one side because if you're just standing around with your horse you just pull the reins from over his head and hang onto them that way. Hah but that's neat I didnt know they made reins like that!
Yep, they are the exact same thing as most leather roping or sport reins, just a little flashier. :) As for the snap, it makes tying really handy. It can get tricky to try to tie a roping rein to a post when it is still doubled and attached on both sides. It also helps if you ever let your horse graze a bit with the bridle on. I have had a horse step through the reins with a front foot and break them when they came tight.
The speed events at shows where I ride (tiny, local schooling shows, mostly) require that riders dismount and lead their horses out of the arena after their run. I kind of figured that that's why some barrel reins had the snap on one side, to lead easier after running the pattern.
I love my nylon barrel reins, though. Easy to handle and I don't have to worry about dropping one split rein on a trail ride. Mine have snaps on both ends, so they double as a lead for my trail bridle.