I had a breakthrough of sorts this weekend when I rode Louie. I had had a lesson 3 days prior, and I really focused on what we worked on that day- controlling every step, half halting in the corners and turning on the forehand if he didn't respond. All the while, working on this square exercise that's perplexed me for 2 years, I think I worked it out for myself during this ride.
As a reminder, the square exercise, according to my original understanding, was an exercise performed at any gait, where you basically ride in a square, asking for a half halt followed by a turn on the forehand at each corner. Some how this is supposed to make the horse half halt and use their hind end with each corner.
So here is my basic interpretation/concepts that help me to understand how to use it during my ride... the square exercise is basically the preliminary step to install a "half halt button" on the horse. when you ask for a half halt, if you don't get an immediate answer, you briefly apply your inside aids (whip, leg, rein) as a correction to say, "hey, wake up, I said half halt/slow down, and you just blew through my aids." It has to be a quick/abrupt correction, not just pulling the horse in a circle, but getting their front end to stop abruptly while any bit of their remaining momentum is carried out by the hind legs until they come to a near-halt, then you can proceed forward and relax the aids and carry on.
If you do it frequently- like 4 times in every circle, they come to expect that they could be asked to stop at any moment and they start moving about under much more control. After a few corrections, they start to feel those inside aids when you start to ask for your half halt (core becoming still, inside leg (well both legs, but more inside), inside rein) and they start to automatically half halt because they know the correction is coming if they don't slow down, rebalance, and stay with you. Eventually, you can go around and use a more subtle core contraction/stilling of the seat with a little bit of leg (seems that applying more inside leg gets the desired response) and closing the inside hand just slightly and they offer the half halt with much less drama. So basically, the correction (quick turn on the forehand) results from an exaggerated form of the cue for the half halt.
I worked on this at the walk for a long time, then the trot for a while until he seemed really responsive on both sides, then I tried it at the canter (only half halting about 3 times per circle instead of my usual 4- it's just tough to do it every other stride for starters) and we had some really great half halts! Especially at the canter, he felt the inside leg and just rebalanced and slowed down beautifully. I then used it to do my downward transition, and while it wasn't great, it was a lot less rushy/trotting/falling down than normal.
So, I'm not sure if I've got the right interpretation of the exercise, but after 2 years I think I finally understand it. The square exercise installs the "half halt button" by showing them a very mild form of the same aids (inside leg, core, inside rein) that would be used to quickly stop their front end in the turn on the forehand.
not sure if it's right, but worked well for us! Success!