Almost every swing motion you are rotating and counter-rotating around your center of mass, which is around your belly button. Its not immediately obvious that when you hit a forehand that your legs push in the opposite rotation of your swing, but if you try to hit a forehand or backhand standing on one foot, the foot in the air should push in the opposite direction. ie if youre right handed hitting a forehand standing on your right foot, your left leg should kick forward. If standing on your left, your right leg should kick back because thats opposite the rotation in relation to your swinging arm and your center of mass. You never want your whole body to be rotating on a swing because it sacrifices so much control.

Same goes for the serve but its a vertical to horizontal rotation, the type of rotation if you were to jump up with your hands straight up then try to touch your toes. With the serve you do want your whole body rotating forward because it gives you more power but not enough to where you would just fall flat on your face. Now the leg kick comes after the service motion when the player needs to land and get in position. If you were to do this just jumping straight up then kicking one leg back, your torso would rotate backwards a little bit but because youre already bent over after serving, the kick back just helps to land a little more vertical than you would without