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First some background. The technical center of Tomiki's teaching system is Junana Hon Kata (the seventeen fundamental forms). This set of techniques is divided into 4 parts - atemiwaza, hijiwaza, tekubiwaza, and ukiwaza. Kotegaeshi is a tekubiwaza (wrist technique).
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In all of these techniques, tori is trying to establish connection to and control of the movement of uke's center of mass. In the atemiwaza, this is done by making a connection directly to uke's centerline and applying a spinelock so that tori can move uke's center. In the hijiwaza, tori is connected to and moving around uke's elbow (but still trying to get to uke's center). In tekubiwaza, tori is hooked to uke's forearm and is moving around (still trying to get to uke's center). In the ukiwaza, tori is hooked to uke's hand (still trying to get to his center).
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So in each successive part of the kata there are more degrees of freedom between tori and uke's center. More technical skill is required to control the slack between the partners.
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In kotegaeshi (all forms of it), tori is connected to uke's wrist/forearm and is moving around through space around uke's wrist. Tori has the same difficulty here that he has in all the techniques - controlling the slack between uke's and tori's centers. If tori just pushes on uke's arm any old way, uke has slack in his wrist and elbow and shoulder and spine that he can use to spoil the technique.
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So, tori turns uke's wrist (or follows it as it turns) until it is at the end of its range of motion. This takes the slack out of it like twisting a towel makes it stiffer. This allows tori to push or pull uke down.
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Another way to look at it. If you watch uke walking relaxedly and smoothly across the mat, you will notice that all of his joints are moving smoothly, opening and closing in a finely coordinated way to allow smooth, balanced motion. But if you flex uke's wrist and turn his forearm until all the slack is out of his forearm, elbow, and shoulder (even if you don't push the joints to pain) then uke can no longer walk smoothly. He has a hitch in his gait caused by the restriction on one side of his body.
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Usually uke can compensate for this restriction over a distance by distorting his posture and sort of limping along. Often you will see him lift his opposite hip to try to get some more control of the assymetry, and this results in a really noticeable limp. But what happens when uke is sailing along all joints working fine and then BAM! all of a sudden there is no motion in one side of his body and one leg is suddenly shorter?
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I usually think about kotegaeshi as a technique for removing all the slack from one side of uke's body at the moment of a footfall. When this happens, uke usually reflexes into a severe offbalance and perhaps even a fall. Even if uke doesn't fall immediately it is usually easy to push him down with your next step.
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So I think you're on the wrong track if you are trying to look at the superficial commonalities (whether you are flexing or turning more, the direction, whether you are pushing or pulling or doing guruma or otoshi). What is common among all these techniques is...
- move in synch with uke
- watch for a footfall
- on that footfall, take all slack out of one side of uke
- push/pull uke to the ground however you have to to get him down
(N.B. The same recipe works great for kotehineri too)
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Now there are instances of kotegaeshi that don't exactly fit this recipe, but this is how I usually think about making that technique work and I have gotten good mileage out of this.
