Straightness is an important goal in any performance horse, but there are degrees of straightness and a lot of confusion about how it is defined. Basically, it means that the hind end should be in line with the front end, not just on straight lines but also curved on the same arc on a curved line.
Obviously, this shouldn’t be the rigid "straightness" of a 2-by-4 piece of lumber. The rib cage area should always be accessible to the rider's inside leg in order to lift the rib cage and keep the horse’s topline supple. In order to do this, it’s useful to think of the head/neck and body assemblies as separate units. You should be able to softly bend the neck to either direction without the body slipping sideways.
Of course, that is usually easier said than done, as anyone sitting horse that is shying knows full well. When your horse is horrified by something to his right, his head goes right while the rest of him goes left. That’s if you don’t have control of the outside shoulder, of course, which could prevent the shy in the first place. The basic tenet is that the body of a horse tends to slide away from the the direction of the head/neck. However, a rider doesn't want that to happen when he's trying to perform a line of jumps or flying changes, which is where training for straightness comes in very handy.
I have been concentrating on straightness a lot more lately, since I am trying to get better with one-tempi changes, my personal bugaboo. (One-tempis are flying changes every stride, from 7 to 15 in FEI dressage tests.) I can keep a line of twos straight without any problem, but the ones are a bear for me.
I also see issues with straightness demonstrated very dramatically when I judge a dressage test that calls for a shoulder-in on the center line. Without the aid of the rail, where a shoulder-in is usually performed, the horse often wobbles all over the place. It should be an "aha" moment for a dressage rider if they can't get a good score for that movement.
Here’s an interesting test for straightness that my coach Ashley gave me recently: While riding in a straight line (anywhere, on the ring rail, inside the rail or even in an open field), gently direct the horse's nose to one side. Don’t pull or stiffen against the bit, just widen the hand, directing the head not only to the side but also lower if you can. Now, take notice of whether the shoulders or even the entire body slide the other way, which could be subtle or overt. The question you have to ask yourself as the rider is whether you can control the direction of the horse’s body with your legs and seat when the head is directed elsewhere. If you can’t, then you have more of an issue with straightness that you realized, because the test has indicated that your only real control aids are the reins.
Not only is this a test but also a good suppling exercise. You should be able to soften the neck gently to one side while riding the body straight at all three gaits and on both curved and straight lines. It’s easiest at the trot, because the feet can become planted at the walk and your horse may need extra strength and balance to counter-bend at the canter. It’s a good exercise for warm-ups/cool-downs and as a break in mid-work, in addition to being a test for the correctness of your work.