What causes disc herniations to be asymmetric?

Annulus weakness: It is believed that there is usually an inherent weakness in the annulus which is the covering of the disc. The actual cause of the weakness is not clear but may be related to stress/injury that has lead to a small tear. This then allows the nucleus pulposus to push out through the weakness thus leading to the "ruptured" left or right sided disc.

Interesting Question: Most likely the answer is that you yourself are not symmetric in any way shape or form. There are asymmetries found in your body and spine all the time and which leads to the disc herniation also being asymmetric. Another common reason is that the ligament that hold the disc in place does not cover the whole disc only the central part, which leads to the disc herniation going to one or the other