I find it helpful when teaching this to get people to focus their eyes' on the geometry they are riding. Front to back balance is something you have to to learn to feel for. I tell my students if your horse has a nice cadence on contact with the reins and then you let the reins go loose, If he speeds up, he is probably out of balance and dependent on your reins for it. Many people think their horse is "quick" or fast and it's just out of balance.
Lateral or left to right balance can be felt really well by practicing a simple exercise. Ride a straight line toward something. In an arena, It could be down center line (dressage letter c or a). In a field or non letter ring, It could be a cone, a fence post, a tree, your dog, husband/wife. Focus your eyes on your target. The first ride don't do anything just feel what you horse is doing. Is he going dead straight or sort of meandering or drifting to one side? Don't do anything yet! Just gather information and say to yourself for example" hmmm precious seems to wobble left and right a lot" or "sugar seems to veer left all the time". Do this a few times until you really feel a pattern. Then start fixing with you legs (reins if necessary). Your horse will now probably over correct or not correct depending on how responsive he is (he should know how to move of your leg sideways for this). Keep your eyes up so you can see if you corrections are on your target line. Or they went passed. Keep making corrections until your horse track dead straight down the line with NO corrections from you. This may take time if you have ridden for a while with out awareness of this. You may find your horse it quite crooked and reluctant to track straight. Eventually you can take this to circular pattern using visualization to see your circle on the ground.