First, find something to talk to them about which has nothing to with the reason why you are rivals. Most of the time rivalry is a result of jealousy, so find something that is not a bone of contention between you.
Second, talk to them about something that they are very good at or interested in. This will make them relax because they have the home court advantage but at the same time they are not competing with you.
Third, when they are not around make sure that you speak about them the same which you would while they were present. The worst thing that you can do to a rival is say something which fuels the fire. Especially if it comes to them through another source.
Fourth, make sure that you are not too nice to them when you approach them. If you are they may lose respect for you or think that you are simply 'up to something.'
Fifth, find something that you have in common which you are both passionate about that has nothing to do with the reason you are rivals. Nothing will unite two people like a battle so if it is a cause to fight for, all the better. Just be sure that it is not another person because that will not last long.
Sixth, ask for their advice. This will impress them that you are humble enough to ask them for advice and that you respect them. Many times rivalry is due to a lack of mutual respect, if you can restore respect you can heal wounds caused by conflict.
Seven, entrust them with something. Just make sure that you have done some of the other ten steps to making friends out of rivals first. Trusting people puts them on the hook to live up to that trust. They will not forget it. Just make sure that you use common sense.
Eight, tell them a joke. Seriously, making someone laugh is the best way to get them to relax. If they think that the joke is funny they may even try to one up you with one of their own. Just make sure that you laugh so that it will not be competitive.
Nine, as the friendship grows ask for them to forgive you for being the way that you have with them. They will most likely follow suit.
Ten, Find something productive to do with the friendship and it will probably last a very long time. So there are your ten steps to making friends out of rivals, now you are ready to turn that bad relationship around and win a powerful ally.