Eliza,
Here are a few thoughts and what I have learned over the past four years about pivots.
There are givens such as proper structure, balance, standing on one's own feet and general good dance posture. Not rigid and not loose. Just a beautiful balanced stance from both partners. Tension is pivot killer.
Number of posts have mentioned tight abs, gluts and so forth; I call it the individual center first, then the center of the partnership. This is not a very static thing as the dancing is not static. Center of the partnership stays consistent throughout the dance and balance is not static, but very dynamic.
One of the most common mistakes men do (as I am a man and guilty as charged) is the mental approach that we can muscle our way into the pivots and take the lady for a ride. As it is possible to do so, it is very unpleasant, too much work and very tiring. Not allowing the lady to do her part, create the rotation and the spinning feeling of the pivot. This means that the ladies need to be very active and know when they need to power through.
With the above in mind, most observers and students think of pivots as rotating steps. The rotating steps are called spins. I never confuse the two. If you think of and treat the pivots as rotating steps, then we are doomed from the start. I think of pivots as linear steps. How else will it travel along the line? And if you take the corner from one wall to the next, partnership will cut the line shorter or longer.
Also most dancers come to the center of the turn to do the pivots and kill the momentum of the movement. We need to respect the circle of the top lines and keep our big heads in their respected positions and not to pull in with our arms as it compromises the integrity of the frames.
Fallaway is simply going back in PP. If we are not comfortable in PP, then the fallaway will not feel right either. In PP, top line, turns more that the hips and head moves in unison with the top line, not more, and not less. If we try to keep our heads too stretched, then physically we are retarding the motions of the top line.
I hope this helps…………..r.