They do look like long lines, but you probably won't see them being used outside classical dressage circles. The book "Training the Horse in Hand: The Classical Iberian Principles" by Alfons J. Dietz goes into the pulley set-up and why it is used. The chapter "Work with Long Reins" starts on page 64:
Sara, I don't think it's the same thing. Your link shows two identical reins and two pulleys on each. This is two different reins, with one pulley only and the funny hook thing on the other without a rope. At least, that's what it looks like to me. I'm so curious about this now...
I still haven't found out what they're used for, and as they're at the stables now, I can't compare them with those mentioned in the book (I have that book gladly, it's a good one).
My trainer is an ex-ecuyér from the Cadre Noir du Saumur, and not even him was able to tell what this thing is used for. It came pre-packaged, so there's very little chance the rein/bridle people messed around with it.