Even the cues under saddle must be taught in both directions, at all the gaits, under different circumstances, but yet consistently the same, for the horse to be confirmed as mastering that cue. Otherwise, another rider will not get the correct response from the same cue because it is not possible that they will perform it exactly as the horse was taught. The horse must be able to recognize that the cue is like one their first trainer taught and choose the correct answer. Sometimes they might pick the wrong one due to the fact that many cues are similar and riders have as many differences in the way they use their hands, leg, and seat as horses have in the way they respond to them. A sensitive, experienced rider will calmly repeat the cue and reward the horse with a pat or "good!" when he responds with the correct answer. If the horse has been trained correctly it is very likely that it will be able to figure out what the new rider is asking.
Just as in humans, some horses learn faster than others, some have shorter attention spans, conformation issues that make the lesson harder or easier for them, differing sensitivities to stimulus, and different physical strength or weakness in different areas of their body. So it is very important that the rider/trainer consider these differences and apply their knowledge to each horse individually. Too often a trainer falls into a "system" that they apply to every horse without recognizing the individual differences in each horse and then declared that a horse is "difficult", "inferior", or "untrainable". A young horse might also be labeled "superior", "talented", or a "star" because they have the mental and emotional make up to excel early in their lessons, but no consideration is made to whether or not they have the conformational or physical development to sustain the level of training and they break down. In many horse sports these talented youngsters appear to be expendable - there are always more coming along - so the long term quality of life is not considered.
Many sports, both human and equine, are striving for the perfect athlete. Perfection is rarely possible, but the ability to recognize individual differences and develop a training program that will address them without force or unrealistic expectations should be the goal of every trainer. I have seen horses and humans who were able to perform at levels that conformationally they appeared not to have the ability for, but someone had brought them along with patience and proper conditioning of the mind, body and spirit, and they easily and happily won the gold!