According to modern physics symmetries in nature always point to energy conservation. This is called the Noether's theorem. Whether one deals with discrete, continuous, gauge, or internal symmetries, the results will be the same... a validation of energy conservation. With this observation Scientific Godism proposes the following reasons behind the validity of natural symmetries:
1. A corollary to the law of Cause and Effect states "nothing can exist, develop or move without the usage of energy".Thus all existing reality is but a form of energy as per Einstein's equation E=Mc^2. And energy, light, and matter are one and the same in diverse forms.
2. Since energy/light/matter can neither be created or destroyed, all existing reality (linked to energy) must be the result of action taken by a higher self-existing being. For this article let us call this high being GOD. Also under the same assumption of energy conservation linked to symmetry, the only way for created beings to properly utilize God's energy is to always ensure harmony and balance in their existence, development and motion. Any imbalance will result in dire consequence as cosmic forces will work behind the scene to restore balance and harmony. Our current concern with global warming is a clear example of cosmic forces working to restore balance to the ecosphere.
3. Thus we can say that nature/God appears to abhors imbalances and will actively seek to correct or eliminate all or any existence, development or motion which deviates from balance and harmony. A simply example of this is the human dislike for ugliness but love for beauty. Ugliness is but a form of imbalance or lack of symmetry.
So in the above ways we can see that symmetry and conservation are the same, an attempt to preserve balance and harmony. Thus we can conclude that symmetry rules in mathematics seek to ensure harmony and balance in the usage of God's energy. Another expression of this law is the concept of "the path of least resistance".
For more details on this concept please refer to my article on consciousness.