What I do is a little different than some...and that is I consider several aspects of a lateral move as separate, rewardable components and will reward them individually here and there, a little randomly. I have said from the outset that I tend to train both ends of the horse and work toward the middle
Josepha is a wonderful teacher and what she says is true...if you reward only for one thing, concentrate on the hind end for now. It is very helpful to have someone watching and let you know how well he is stepping under if you can't feel it for yourself.
I can feel some things better than others! For instance, I am terrible at judging how far my horse is stepping under behind. I can tell when he's stepping under of course, but not the quality of it exactly. So if I have no help, I will reward for fluidity (if it feels smooth and effortless) and really, the horse's state of mind...which is rather funny because he is always in a good frame of mind if I do my part correctly...so he gets the reward when I do it right :-). It's only fair...he puts up with me when I don't get it right!
Another thing I can see/feel more easily is whether or not he is using his inside shoulder properly, and not dropping it. If the shoulder is lifting, it is more likely he is also lifting from the base of the neck (someone correct me quickly if I'm mistaken here!) and it seems that a natural result of this is a nice arching of the neck....and a natural prerequisite is a proper bend. So if he's using his inside shoulder properly (which I can see/feel) then I know that several other things are also going correctly. Reward!
But in groundwork, you can of course see it all happening - so you don't really need a spotter to watch you - then you can work on the ground, reward often for good hind end usage. Then you get on, try it in the saddle and then get off and do it on the ground again. If your balance is ok in the saddle, it's likely he's also doing it quite well then too. So don't be afraid to reward if it happens to just feel good.
But back to my "both ends toward the middle" philosophy....
I will concentrate on, and reward for, correct front end usage in some cases when that has been a difficult thing for the horse to achieve for some reason (which may include isolating and rewarding only for ramener in movement - especially when bridleless!). In Tam's case, he tends to overflex, so I reward him for a more moderate flexing/bending when I see it happen (both on the ground and in riding).
So what components might I isolate and reward independently?
I would love it if everyone would add to this list, because I might miss something good!
1) Stepping under
2) Lifting the inside shoulder
3) Quality of the bend
5) Emotional control (which is more up to me than the horse...it's my responsibility to keep my request clear, understandable and not to ask more than the horse can do...and he will stay calm and relaxed in return...but he can still be rewarded for it!!)
6) I will even reward if I see the abdominal muscles contracting...because that tells that the horse is really lifting into the movement and not just moving sideways.
The horse will tend to know what I'm rewarding for if I a) focus on that part of his body and b) through consistency (they will tend to sort out what body part is being rewarded).
I will reward for any tiny attempt to do things correctly in a way that is healthy for the horse and within his ability at the time. So I will not wait for a full, correct movement. I will reward for any attempt the horse makes in the movement while he's learning.
Eventually, by rewarding for all these independent parts, and also by NOT rewarding occasionally, the horse begins to combine the elements and in essence, begins to form "sentences" out of all the single "words" he's learning. And getting loads of rewards in the process!
I hope that helps! Can others add to this please? Or correct if I've said anything inaccurate (which I am known to do!!!).