Ahhh... the posting trot..... I LOVE it! This is how I teach it:
The horse uses his legs on the diagonal to trot, so at any time there are the two opposite diagonal feet on the ground and the other two are off the ground. Then it reverses.
As the 'power plant' of the horse is in his rear end, it is actually the hind legs that are allowing the forward movement. However it's really hard to see/feel/know what the hind legs are doing, so we use the front legs to teach the movement to the rider as they can look down the shoulder and see what is happening.
But then you must realise that whatever the left hind is doing, the right front is doing. So in order to know where the power of the horse is coming from, and utilise that power, the rider has to look for the opposite in the front end. For example, when travelling around a circle to the left, the left hind leg is doing all the 'pushing' off and providing the power to go forward, which is also the right front.
So when the left hind is hitting the ground, and then pushing off the ground (providing propulsion) the right front is doing the same thing. In order for both the horse and the rider to be in sync, and in order for the rider to utilise the propulsion force of that left hind leg, the rider should be sitting as the horses left hind leg hits the ground, and then rising as the left hind leg leaves the ground. If you consider that the horse's left hind leg is actually 'pushing' you out of the saddle as it pushes down toward the ground to lift the horse off the ground, you will raise up out of the saddle.
Do an exaggerated walk yourself, bending your knees with every stride, and pushing off the ground before putting your other foot down. You can feel the 'lift' that you generate in your body when you straighten your knee to take the next step.
Similarly, as the horse pushes his left hind off the ground, he pushes you out of the saddle. So, as the left hind is coming off the ground, so is the right front leg. If you are sitting as the left hind hits the ground, and rising as the left hind leaves the ground, it follows that you are sitting when the right front hits the ground, and rising when the right front leaves the ground. This is easier to see, as you can look down the shoulder and see when the right front shoulder is stretching forward (right front foot is leaving the ground).
So, in riding a trot in a circle around to the left, you should be sitting when the right front should is moving backwards (which is when the right front foot is hitting the ground). And so it follows that you are then rising when the right front shoulder is moving forwards (which is when the right front foot is leaving the ground), and this is the same time that the left front shoulder is moving backwards because the left front foot is now hitting the ground.
You will end up sitting on the 'outside' front foot, and rising on the 'inside' front foot.
This is what Elizabeth said earlier, "Stand on the inside leg" (which infers, "Sit on the outside leg"). Referring to the FRONT legs.
Practice by trotting about 50 foot circles constantly until you feel the difference. Trot around to the left, sitting on the outside (right) leg, and then trot around to the right, sitting on the outside (left) leg. On the circle you will definately be able to feel the difference between sitting on the outside leg vs sitting on the inside leg. Once you are on the correct diagonal, the horse will feel smoother... it's a case of centrifigal force at work.
Out on the straight trails, the horse will benefit by you changing the leg you are sitting on regularly. The muscles need to be evenly exercised on both sides of the horse's body, and if you only ever rise to the trot on one side, you will be able to see a difference in the size of the muscle when you are looking straight down at the horses' withers from the mounted position. If you can't see the withers because of the saddle, take the saddle off and just sit on your horse bareback and then close your eyes and run your hands across these muscles.
