I think some of you are confusing the term 'dressage' with the sort of thing you are maybe seeing in the competition ring
The first 'book' on dressage actually dates back to 360BC and was written by Xenophon in part of his guide to horse management, it became a military thing where the horses natural ability was used by a rider to produce a horse that could be a partner in battle...
Xenophon didn't write on dressage. He wrote on training a military horse, and how to train a horse to look good on parade. They wore skirts, no underwear and used a spear to help them vault onto their ponies. They used vicious bits, no stirrups and had their feet well forward. He didn't believe in having the thighs parallel to the ground, which is about as close to dressage as Xenophon got.
That mis-history is part of the mythology of dressage. Many of the past 'masters' recommended training programs for horses that would give a heart attack to most modern riders.
Modern dressage has basic levels, not because the basics are dressage, but to make sure the basics are in place before allowing a contestant to attempt real dressage. The basics are not 'dressage', but basics are required for most competition. Jumpers start a rider over a pole lying on the ground. That isn't jumping, but you have to be able to cross a pole before you can move on. In some sports, it is even simpler. In polo, if your horse sucks, you suck and you lose the game.
It is no more wrong for a reiner to ask a dressage rider for ideas in training (and vice-versa) than it was wrong for a green rider with a green horse (me & mine) to ask a barrel racer for ideas on how to loosen him up . That didn't make me a barrel racer. Nor does it mean barrel racing is the basis for all good riding.
I'll be honest. I always thought 'travers' was the name of the actor who played George Adamson in "Born Free"
( ...in WW2, Bill Travers was left behind in Burma. To avoid capture, he disguised himself and walked hundreds of miles behind enemy lines to reach friendly territory. Just an odd note of history from a film buff).
Anyways, I obviously don't know what a travers is or how to do it. But if I understand what the posters here are writing, a reiner uses it to affect how the horse moves from point A to B. Reining is about performing patterns well, and a travers (I don't even know if that is OK English) makes it do so. In dressage, the goal is the horse & the gaits, and the maneuvers are supposed to reveal the horse's inner physical ability and the rider's ability to elicit that capability.
Rider & horse A may nail a circle more precisely than rider & horse B, but if B does it with a horse who is more relaxed and has more self carriage than A, B may get more points.
Remember, I only know these events from seeing videos. I could be entirely wrong. I'm not lecturing anyone on what reining or dressage does, but trying to understand it while we have accomplished riders in both disciplines around.
It isn't wrong to borrow training ideas without taking up a specific discipline. The lady who has done so much to help my spooky mare is from a barrel racing background, which isn't exactly a sport famous for calm horses. The key is to understand that you are borrowing a training tool, not practicing the discipline itself - and I think nrhareiner would agree that she is not performing dressage, but borrowing training tools and/or crosstraining to make her reiners better reiners. Again, if I'm putting words in anyone's mouth, feel free to jump in my chili. I just think we're starting to argue over things we're largely in agreement about...
: After 3 years of riding Mia in arenas only, I've been trying to turn her into a trail horse. We made our first solo ride today. Once we got far enough away that she couldn't hear Trooper screaming like he was dying the death of a thousand cuts, she did well. For a half mile. Alone. At my level of riding, THAT is a big whoopie. I realize it is a yawner for anyone else... That is MY idea of 'basics'!