If you look around the world, you will be instinctively drawn to symmetry. We are hard wired for it it seems. Babies will stare longer at symmetrical faces then non. We unconsciously are drawn to more symmetrical people of the opposite sex. Symmetry to us is beautiful. It's embed in nature. There may be even some deeper underlying health issues with symmetry as men and women with asymmetrical faces are more prone to depression and stomach issues.
In general we know that asymmetry in the body also predispose one to injury. Not necessarily cause injury, but perhaps is a contributing factor, more so then general "inflexibility" or strength.
For instance, if one is on their back and raise one leg up, then the other, compare. The standard is 90 degrees to the floor. If both are about equal, even if they are much less then the standard, the new way of thinking is that it is probably less of a deal then if one leg is more limited. The limited leg may be a problem. I stress MAY be.
There are a few instances where asymmetry may be appropriate such as a baseball pitchers elbow. The lay back position has increased as thousands of repetitions have brought about appropriate bone and soft tissue changes to allow that 90 mile per hour fastball.
Most of us are not in the asymmetry will help category though. Everyone now realizes that mobility is important, but many don't really know where to begin. Movement screens have become very popular as of late for that simple reason. It's easy to understand this is different then this. Asymmetry is not just range of motion though, it can also be a firing pattern. Sure your thoracic rotation was equal, but you were strong on the right and weak on the left. This too is asymmetry.
Instead of blindly doing a half ass warm up at the gym or before a run, do a few symmetry checks, improve them and then go about the workout. Not only will the workout most likely go better, but you may have increased your chances of staying injury free.
Everyone has a trademarked (number with cool sounding name) so I'm calling these the T.O.P. 6. (get it?)
1. On the floor, legs straight. Press one heel into the floor, keep the other leg straight and raise it up. Check the angle and feel. Test the other leg. Posterior Chain
2. Same position. Raise the leg up and slowly abduct the leg away from the body. Keep your pelvis on the floor. Return and test other leg. Medial Chain
3. Same position. Bend one leg, heel on the floor. Other leg is straight. Keep you ankle in dorsiflexion on the straight leg. Rotate the leg internally and externally. Compare to the other leg. Hip Rotation.
4. Same position, bend one leg and keep the other leg straight. Press through the heel of the bent leg and raise hips off the floor. Keep your pelvis level, don't let the pelvis rotate. Feel the glute or did you feel the hamstring? Did your pelvis rotate? Compare to the other side. You want to be able to feel the glute and control the pelvis. Glute Activity and Oblique Strength.
5. Both legs bent. Heels on the floor. Keep your ribcage down, or parallel to the floor. Keep the arm straight and raise it above you to try to get it the hand to the floor. Again keep the ribs down, you will try to hyper extend the low back to cheat. Keep it flat. Test other side. Thoracic Extension
6. Same position. Arms abducted ninety degrees to the body. Bend 90 degrees at the elbow. Rotate the shoulders internally, back to vertical and then externally rotate. Shoulder Rotation
Run through these as part of your warm up. It's feel as much as function. Gain some symmetry. Work for symmetry.