There's a reason the symbol of Christianity is the cross. The cross represents one of the most important aspects of our faith: Forgiveness from God. Through the cross, we who deserve to be condemned instead receive mercy from the Lord, as our punishment goes to none other than His own Son. Through His resurrection, we not only receive forgiveness for our sins, but are actually granted a reward for His righteousness.
From this, we see that our entire religion is based around the concept of reconciliation. Reconciliation, being the strongest form of love, forgets all wrong doing, and actually strengthens the relationship in ways it was never strong before.
If we are to be image bearers of God (e.g. 1 Corinthians 15:48-49), it should not surprise us that He would expect us to do likewise. As John MacArthur put it, "you are never more like God than when you forgive." And so Jesus says listen, before you do any rituals, before you give any gifts, before you do any of that, you make sure you have done everything in your power to make things right with your brethren. The only thing more important than your relationship with others is your relationship with God Himself.
This concept is so strong that Jesus actually links the command to "love your neighbour as yourself" to the command "You shall love the Lord your God with all your heart, with all your soul, and with all your mind." (Matthew 22:36-40). The two commands are inseparable! You can't love God without loving your neighbour, and according to 1 Corinthians 13:5, love does not keep a record of wrong doing.
Thus there is no religious duty or ritual you can perform that is more important than reconciliation. If there is anything between you and another Christian, it is imperative that you do everything within your power to mend that damage. Anything less is of the devil.