All joints maintain stability during movement by activating muscles that pull it in the desired direction while synchronously releasing the opposing muscles. But maximum stability is achieved in a joint when both sets of muscles contract at once to hold the joint in its neutral position. Although the concept of opposing muscles stabilizing the wrist are well known, the concept of stabilization groupings is not well understood in other parts of the body.

Wrist – Flexors & Extensors
To flex the wrist, we activate the flexors on the front of the arm and synchronously release the extensors on the back of the arm. But maximum stability is reached when the flexors and extensors contract together to hold the wrist motionless in its neutral position (straight).

Hip – Nutators & Counternutators
During gait, we alternately contract nutators and counternutators on each side of the pelvis synchronously. During stance phase, the nutators are active. During swing phase, the counternutators are active. Both muscle groups function to maintain balanced tension in the sacroiliac joint as we move.

But, while raising from a stooped position, movement occurs mainly in the lumbar spine and hips, while the sacroiliac joint must remain motionless in order to keep stress off the SIJ ligaments. To do this, both the nutators, e.g., the hamstrings and the counternutators, e.g., the quadriceps, contract together.

Interestingly, while the hamstrings are nutators, there is a slip of muscle coming off the long head of the biceps femoris that connects to the sacrotuberous ligament, which is a counternutator. Wingerden[1] noted that this slip induces counternutation by pulling the sacral apex anteriorly, and theorized that it is part of the mechanism that helps provide stability and extra force to stabilize the SIJ by keeping the SIJ motionless while raising from a position of hip flexion.

Internal Abdominal Pressure (IAP)
To achieve spinal stability, in which the trunk below the diaphragm remains relatively motionless, it is known that the Transverse Abdominis tenses the front, the Multifidus tenses the back, the Diaphragm tenses the superior part, and the Pelvic Floor muscles tense the inferior part.

In normal movement, such as gait, the nutators and counternutators reciprocate side to side. Also, during inspiration, the counternutators are active, while during expiration, the nutators are active. But, during spinal stabilization, both sets of muscles, nutators and counternutators, contract simultaneously to stop spinal movement below the diaphragm, and increase IAP.

Although increasing IAP may be done at full inspiration, it is usually associated with expiration. Because increased IAP is a function of the four muscle groups mentioned above acting as a unit, it is generally assumed that they all function in the same phase, either inspiration or expiration. Without considering the opposing action of stabilization groupings, and the mistaken idea that all muscles increasing IAP contract at the same time, it is thought that they are all expiration muscles and, by extension, nutators.

Further, in an intra-anal experiment, because the piriformis muscle contracted during expiration and relatively relaxed during inspiration, it was again assumed that the piriformis is an expiration muscle. However, the investigator did not consider the concept that these muscles are part of a stabilization group, as described in the Musculoskeletal Integration Theory. Since stabilization groupings must include muscles with opposing actions, it would indicate that the piriformis, opposing the Transverse Abdominis (a known expiration muscle), is therefore an inspiration muscle and hence a counternutator in gait and other movements.
Additionally, if you look at the line of drive of the piriformis, it can be seen to be a counternutation muscle.

Consistent with other joint stabilization muscles, the piriformis contracts as a counternutator during movement but also acts as a nutator to hold the trunk motionless during expiration and increasing IAP. (See Influences on SIJ Movement).

Here, we have another example of the relative importance of looking at the musculoskeletal system as a function of the opposing actions of nutation and counternutation.

References:
1. Wingerden, J.v., A functional-anatomical approach to the spine-pelvic mechanism: interaction between the biceps femoris muscle and the sacrotuberous ligament. European Spine Journal, 1993. 2: p. 140-144.