I find it interesting that while there are many kinds of koshinage, or hip throw, most aikido dojos only irregularly practice the technique, and then just one particular type and its variations. For example, in Iwama dojos the “duck under the arm” yonkyo-grip koshi is the preferred flavor, with attention given to a twisted crouch while looking up at the ceiling before and after the throw as nage’s shoulders rotate. In Birankai it seems to be the “step in front” arm-around-the-waist koshi, usually practiced as henkawaza from a failed yikkyo.
My last teacher made the useful distinction between “head-under” and “hip-under” entries. In the first, like the Iwama variation described above, nage’s upper body moves in front uf uke who is stretched across nage’s lower back by an extended arm. In the latter, like the Birankai variation, nage moves his lower body into position first and leverages against uke’s lower back. Both throws need hips low and in contact with uke’s center to be effective. Both are most effective when uke’s center (the knot on uke’s belt) is positioned at or slightly “above” nage’s spine (since nage is bending), making the fulcrum just below uke’s center of gravity. Apart from these conditions, the head-under variation often fails because uke falls to high on nage’s back, making it difficult to raise them up and over. The hip-under variation often fails because uke is too low, and can easily roll off to the side.
Variations to each include adjustments to the holds: the head-under can be done by gripping the fabric of uke’s sleeve or from a partial shihonage, and the head-under can have an arm wrapped up around uke’s shoulder or even around the neck instead of the hips as leverage. I particularly like the “sideways double yonkyo” head-under variation from ryotedori, but it’s practically indescribable. Go find your own.
There is at least a third major variation. That is the “ushiro” koshi, where uke and nage are back to back, and uke flies over face-up. This can be done at 90 degrees by grabbing uke’s collar after entering as in iriminage, at 45 degrees with a hip-under by reaching around uke’s neck, or straight over the top with shoulders entwined. I’ll just leave those to the young folks.