Rorty's ideas are deeply religious, without making use of religion or religious vocabulary. In other words, part of what Contingency does is to explain a type of alternate morality that can replace more dogmatic types of morality, like the ones that come from fundamentalist Christianity. Instead of making rules for behavior, though, Rorty makes rules for self perception.
The first rule in Rorty's toolbox is that no one should forget that every aspect of our perception of reality is limited and subjective. That makes our points of view "contingent" on our (potentially misguided) beliefs. In other words, Rorty says that humans are not capable of obtaining objective knowledge about reality or human nature, but they can be flexible and humble. So rule number one in Rorty's non-religious religion would be: Don't forget your perspective is probably wrong.
The second rule is like the first: Don't forget to have a sense of humor about yourself. Rorty explains this axiom in light of irony, the strange way that sometimes things are different that what we believe about them. If this is true about our perception of the world (and it is; for instance: we can't see atoms, but they're real, so reality is different from our perception, because our sight is limited), then we should also remember that our 'self' could surprise us. Therefore, when defining ourselves, we ought to use a pencil, not a pen, because we will make mistakes in our self-perception.
Thirdly, Rorty explains that humans are wired to believe they are kind and generous, but few humans actually learn how to become those things. In other words, we often ignore the aspects of our personality that are cruel and contemptuous. Rorty explains that this is a default mode which is cruel and misguided, and he urges his audience to move forward in their development by choosing to sacrifice their limited, selfish interests for an interest in the greater good. He says our conception of "we" should be stretched until we feel intimately connected to all humans.
These ideas (open-mindedness, self-sacrifice, serving a higher good) might seem Christian in flavor, but Rorty manages to explain the truth of these claims without appealing to religion.
