I find it difficult to have friends whose basic values are in opposition to mine. There's a large space within a person's value-set in which friends can have widely different views, but for a person to have an opposing set of values to mine I don't think is possible. That's not to say that they can't have different values, but if those values are in opposition to mine, I don't see much space for friendship.
I say this because my values are of great importance to me. I value fairness; I value equality; I value honesty; I value life; I value forgiveness. Now, within all these different, basic values, there's a huge space for disagreement on how they're interpreted. I say that I value life, and while another friend may also value it, we may differ on what the context of that value means. But if a person did not value life, or he did not value honesty, I don't see where a friendship can thrive. After all, my friends are my friends because they share my same values.
Let me give another example. For several years I had a friend in college who was, and is, a staunch Republican. He's also a brother in my fraternity. We got along well for several years, until I realized that he didn't share many of the same values that I shared, in particular the value of equality. I had gone through a period of intense scrutiny of my values, and their importance to my life, and after realizing that this friend did not, and would never, truly value me as an equal person, and therefore didn't value equality, I realized that we could no longer be friends.
I think people must adhere strongly to their values, though always with an open mind that accepts that they may be wrong in valuing a certain thing; but, until someone with an opposing value to mine is able to convince me of the rightness of her value, I would find it hard to be friends with her.