Thanks for the excellent questions. I find this is an area that is vaguely understood, at best, by the average horse person. First of all, as far as the difference in the rein aids between English and Western, to me there are none. The rein aids work the same and the horse responds the same way no matter what style of saddle you ride in. Some might argue that the neck rein is strictly Western, but I like my English horses to know the neck rein too and it is imperative for sports like polo (which may be considered an English discipline, since it is done in an English saddle but with one hand on the reins). All of the other rein aids, direct, leading/opening and indirect are definitely used both English and Western.
The term "rein aid" refers simply to how the rider moves her hand and the direction of pull on the horse's mouth (up, back, sideways). The term "rein of opposition" is sort of an old-fashioned term and is most often used with the term "direct rein," as in "direct rein of opposition." Opposition refers to the forward motion of the horse and whenever you pull back on a rein, you are pulling in opposition to the horse's forward movement. Therefore, it tends to slow the horse down.
For the direct rein, the rider's hand moves from the regular hand position (in front of the pommel, straight line from rider's elbow to the corrner of the horse's mouth), directly toward the rider's hip. There is a backward (and slightly upward) pull on the rein and therefore it is a rein of opposition.
An opening rein or leading rein is when the rider moves her forearm to the side and not back and therefore it does not inhibit forward motion. This rein aid is often used as a training rein aid, such as when you are first teaching colts to turn or when you are teaching a horse to spin or turn on the haunches or do lateral movements. It is a leading rein when it is the inside rein (you are opening the rein on the same side as you want the horse to turn). It is an opening rein when you are using it as the outside rein, when the horse is bending away from the opening rein, but you want to move the horse's shoulder or barrel out (like opening up a circle or leg yielding/two tracking).
There are two indirect rein aids: the "indirect rein in front of the withers" and the "indirect rein behind the withers." The latter is a rein of opposition and the former is not. The indirect rein in front of the withers is a lift up and in on the rein toward the horse's neck (an upward diagonal pull on the rein; from the normal hand position, just turn your pinkie toward the horse's withers without pulling back; the inside rein comes across the horse's neck in front of the withers). The indirect rein in front of the withers moves the horse's shoulder in the opposite direction, while the nose stays bent in the direction of the turn.
The indirect rein behind the withers has some opposition or backward pull, and causes the horse to move his hip away from the rein hand while the horse stays bent toward the rein hand, such as in a turn on the forhand or disengagement of the hindquarters. The direction of pull on the rein is up and back toward the rider's opposite shoulder, in a motion like crossing your heart (the inside rein comes across the horse's neck behind the withers).
Some important caveats for all rein aids: it is not the amount of pull or contact that causes a reaction in the horse, but the direction of the pressure on the horse's mouth or the movement of the rider's hand (when using the indirect rein aids especially- it is only effective when there is little or no pressure on the horse's mouth). Also, when riding two-handed (as all of the above rein aids require) your hand should never cross the horse's withers. If it does, the rein aid you are using is ineffective and may be interfering with the horse's motion (pulling his nose in the wrong direction). All rein aids are supported by leg aids (but that is a whole other subject).
The neck rein is typically used for one-handed riding, but may be used two-handed in combination with another rein aid. For example, when you are teaching a young horse to neck rein, you may use the neck rein as the outside rein aid and the leading rein on the inside to help control the horse's nose. Eventually, the horse associates the neck rein with turning his neck and nose away from the rein and you no longer need the leading rein.
Like the indirect rein, the neck rein may be used in opposition or not. The basic neck rein is a gentle touch of the rein against the side of the horse's neck well in front of the withers and has no opposition. The horse is trained to move away from the touch of the rein on his neck and he moves his nose and neck away from the neck rein. If there is a hard pull or the rider's hand crosses too far over the midline of the horse's neck, it will inhibit the horse's movement and turn his nose the wrong way.
The neck rein with opposition (a slight backward pull with the application of the neck rein) is called the "bearing rein" and may be used to turn the horse back on his haunches, such as in a roll back or a pivot on the haunches.
This is a lot of information about how to use the reins effectively and it takes a lot of time and experience before the rider is able to use the rein aids so explicitly and effectively. And it never ceases to amaze me how responsive a horse can be to the lightest amount of pressure and the slightest movement of your hand. One really important thing I have learned through the years about rein aids is that the slower you move your hands, the better the horse will respond.
Trainer and Clinician
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