Take gyakugamae ate, for instance. For many years, as I was just starting out, most of the folks I trained with treated this technique as an "eye threat", meaning I (as uke) am supposed be startled when I see your hand coming toward my face and I should just jump out of my own boxers and land on my head.
Don't get me wrong, an eye threat is definitely possible there. I've certainly had it done to me enough times for real to know that when it works, it works quite well. The problem for me was, it didn't always work, and not just because of inexperience. Sometimes, you get an uke who is built like a fire hydrant and they just don't have that keen of a jump reflex; or, despite years of training, you simply botch the timing. Then what do you do? Wave at they guy?
That's when I noticed someone doing it a little differently. I would randori with Lowry Sensei as a brown belt and let me tell you something, there was no standing up with that gyakugamae ate. No sir, he pealed my head back like a Pez dispenser; I thought little candies were going to shoot out of my neck.
At first I thought, Oh, you may just have to push on the guy's head to get him to go down (if the eye threat doesn't work out). But eventually, I started to see, both in aikido and judo, and in a whole host of techniques, what was really happening: tori was simply lifting my chin.
In fact, you can do a little static experiment. Have uke stand perfectly still and straight. His job is to be strong and not to let you push him down. Then try pushing on his face, from the front or from the side, like shomen ate or gyakugamae ate; try aiki nage/irimi nage, or sayunage. He's a statue, solid as a rock! He may stumble a few steps, but he probably won't be falling down any time soon.
Now try lifting his chin up and then stepping through him. It's an interesting difference. Changing that one element breaks his entire posture!
I know, I know, if you catch the guy while he's moving with, say, a shomen ate, and you stop his head while his feet keep moving, he'll fall. I'm not saying it isn't possible to knock the guy down with the right timing, etc. regardless of the chin, but lifting the chin seems to crumple his posture like nothing else. What used to be made of brick suddenly crumbles like sand.
Now, there is a caveat. The neck muscles are pretty weak compared to other large muscle groups. When you think about it, you can push a stalled car with an "unbendable arm, bridging from the back leg" type of structure. That's a lot of power! Unleash that on a poor guy's neck and, well, you'll send him flying (I've seen it done!) and he'll have a sore neck for days (I know, because I've had one). So we have to be careful, and light. But we need to know it's there, if and when we really need it.
But it also occurred to me that moving the chin to one side or another helps in judo ne waza, too, say when holding kami shiho gatame (it limits his power to only one side) or when choking a guy (push his chin to one side to expose his neck), or when holding yoko shiho gatame or mune gatame, when our arm under his head makes it crook over to one side. The initial balance break from aiki nage/irimi nage does the same thing. Oh, and ushiro kudi gatame....
Basically, just getting the guy's head out of alignment with his body really messes the guy up.
Of course, this is probably old news to most folks who read this, but it's been eye-opening for me to see just exactly how often it pops up. Especially with this action, which I've really enjoyed finding all sorts of opportunities for (the sort of palm up, sideways entry idea):