Have you ever looked for justice or an apology?
Have you sympathized with a cause?
Or maybe used empathy to understand another person's perspective?
Have you employed the Golden Rule or have you showed someone respect?
Symmetry is useful when understanding how people relate to one another: one action is balanced by another which creates equilibrium. This relationship is easy to see in architecture, art, geometry, or science where it is often used as an organizing principle. One side directly corresponds with the other side by looking the same, as in a mirror image.
Now think about how you relate to others.
If someone does something for you, do you reciprocate and do something for them? It could be now or maybe later. If so, you have created symmetry and balance in your relationship with that person. Have you been invited to someone's house and you are inclined to bring a gift? Has someone picked up the tab for your lunch and you tell them that you'll pick it up next time? When you understand that for every action, there should be an equal and opposite reaction, then you are taking a symmetrical view of the world.
This outlook is especially useful when seeking to maintain order and civility in relationships. Achieving balance may not be possible when dealing with multiple people that bring with them different motivations or ideas about how to treat others. When an imbalance has been created, then the rules for how to work with others changes.
One now needs to not only work from multiple perspectives, but also understand how to create balance among multiple parties.