(Yay first post)
This post covers various parts of time. So it is a small history lesson (please don't kill me!)
According to Oda-san Zoro is based on the famous French pirate Francois l'Ollonais. (You can look him up in Wikipedia) To my beliefs Mihawk is based of the English Seadog Sir Francis Drake. I say this for 3 reasons. One: Drake was a legal pirate and worked for the queen. Like how Mihawk kind of works for the Marines. Two: The English are deeply involved in Religion. (Mihawk's sword) and three: Drake sounds a lot Like Mihawks First name (Dracula).
Now; the reason for rivalry is this.
In Europe England and France did not get along very well. I say Oda made the two rivals because he has heard of the wars England and France has been in For example the French and Indian war. So it occurred to him. If he was going to put the believed English Mihawk against anyone it should be the French Zoro.
Does it make sense?
<_<