Pivots can occur in the down position, where the pivot foot on the floor is bent 90 degrees at the toes, angle, and knee with the knee almost touching the floor. Most players lean forward with the head because of lack of strength and are in a very unbalanced position for making any kind of pivot. Pivots also can occur in the up position, where the player has the knees slightly flexed and are on the balls of their feet very much like the triple threat position. A good pivoter can move as easily in either of these two positions and from up to down and down to up position quickly without losing their balance. If you have lost your balance, then correct it before you try pivoting again.
Young players tend to start their pivot before correcting their position.
When you pivot, the hip of the pivoting leg should be pulled through on a straight line to the new position. We teach our players to penetrate to the basket in a straight line, so why would we teach the same players to pivot any portion of their body in a circular fashion. This is not so easily seen, but when they've lost their balance, you can almost be assured that it is because the hips movement did not start in a linear direction and the body was pulled off balance. When I'm teaching my players to go from the up pivoting position, to a down position, where they have shifted the ball to the weakside hand away from the defensive pressure, they must pull that hip down as quickly and straight as possible to the new position. It is as if there is a rope attached to their hip, and someone yanks that rope.