Twists release and return the spine to neutral alignment after deep forward bends and backbends.
Twists affect the internal organs of the torso through what B. K. S. Iyengar calls a "squeeze and soak" action. The theory is that twists cleanse the internal organs in much the same way that a sponge discharges dirty water when squeezed and can then absorb fresh water and expand again. The idea is that, when you twist, you create a similar wringing action, removing stale blood and allowing a freshly oxygenated supply to flow in.
Twists keep you long. As a result of the effects of gravity and the aging process, your vertebral disks get compressed and tend to lose moisture. The "squeeze and soak" theory described above also applies to disks, keeping them plump and healthy, and keeping you nice and tall.
Twists tend to tone the obliques, which are the diagonal muscles in the abdomen that contract to help you rotate. When your obliques get tight, they can pull the lower ribs and pelvis toward each other, which leads to poor posture. Long, strong obliques, however, contribute to good posture.
Twists can teach you to breathe under all circumstances. The breathing pattern is crucial to achieving a healthy twist, with the inhalation always used to establish length along the spine before the exhalation is used to rotate. Sometimes students unconsciously hold their breath when they twist. Learning to breathe correctly as you practice these poses will serve you on the mat and off, making your twists more spacious and imprinting the useful life habit of breathing expansively even under challenging circumstances.
Twists energize and release tension. As you rotate, see if you don't notice feelings of liveliness and vitality. The "ahhhh" moment after you come out of a twist is the psychological version of what happens physically. First you coil up, then you let go.