If we understand the need to balance Compassion with Equanimity, it will help us not expect that we have the power to control everything, which definitely helps us with our relationship with ourselves. We learn how to be more objective with our body and mind. Understanding that it follows according to the Law of Cause and Effect. We learn how to be more objective with other people, understanding they are as they are, and we can't always, no matter how much we care and wish to, help them, as they need to help themselves. This is because if we understand the Four Noble Truths, we understand that everybody needs to let go of the causes of suffering within themselves.
Equanimity strengthens all the Unselfish Emotions, but it goes even further. It helps with letting go of ill-will towards others. It helps us to let go of aversion towards the way things are, especially the inability to control nature. It helps us understand that things are impermanent and we don't have the power to change everything, and because they are impermanent they are unsatisfactory and we don't have the power to change this either. Furthermore, because they are impermanent and unsatisfactory, they are not really mine, and we don't have the power to change this either.
So we begin to see what we can do and what we can't do.
Another way Compassion and Equanimity help us in our relationship to ourselves is by learning how to care about things that are really important and to not care about things that aren't so important. Learning how to be more discriminating in our caring because there are many things we would like to be able to do and many things we want to do, but if we're able to have more wisdom in our life, we see that some of the things that we want to do aren't really that important. One way to understand this within ourselves is to reflect on death. So that we see whether something that we're really caring a lot about, and perhaps even getting upset about isn't really that important - reflecting that we may die tonight.
This helps us understand what is important to ourselves. Encouraging us to care more about the things that are important, like doing your formal practice, taking care of yourself, trying to be more aware during the day so that you can protect yourself and by protecting yourself, you help to protect others.