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The effect of the rein is a matter of degree and of timing in the training.
When we teach the young horses, we insist on turning before bending, so the horse learns first to put weight on the inside shoulder on demand. We do that by really shortening the inside rein and act on it toward 2:00 in a right turn and 10:00 in a left turn. This is supported by a showing the whip toward the horse’s face on the outside or a tap from outside leg. It is really important that the horse first learn to turn without falling outward in any way. Spirals on the lunge are a good preparation for that.
Once they know how to turn, we start to use the weight to load the outside shoulder of the turn and start the bend. The horse is asked to turn left with weight on left shoulder, then release the left rein to start turning right while the weight remains on the left shoulder. All things being equal (the horse being reasonably symmetrical), the right rein can be used fixed to the withers (without being shortened) or open to intermediate/neutral position (3:00). If the hand acts further forward (2:00), the inside shoulder will turn in. If the hand acts further back (4:00 or 5:00) the action will affect the haunches and make then deviate outward (SI position). So this is a delicate thing to do.
Opening an inside rein can be employed (and the withers will rotate to the outside) and the weight will shift slightly to the outside fore, thus freeing up the inside fore. We pursue this until we get correct figure 8s and serpentines (in 3 gaits), but in the above progression I describe because I want to be sure I can put their weight on either shoulder (or right in the middle) as I please. This works very well in trot and in canter.
In the very small circle performed with long walk strides (very different than when done with shorter strides that are less challenging for the balance) there is a natural tendency to push outward with the front legs that must be counteracted by the position of the seat and some control of the inside rein. This is why in that case I prefer the fixed hand inside with release of the outside rein because it does not shorten the neck in anyway (and hardly bends it).