Institutional Church and the Left/Right Divide

I want to pose two questions that have been bubbling up in my mind this spring.  As the Oklahoma legislature debates abortion and immigration issues and as the state of Oklahoma continues to practice the death penalty, it seems to me that the Catholic Church is the only religious institution consistently bridging the left/right divide, advocating against harsh immigration measures, for greater fetal protection, and against the death penalty.

1.  Is my perception correct?  Is the Catholic Church the only (or at least the only large) religious institution that consistently takes positions on both sides of America's ideological divide.  If not, what are the other ones?

2.  If my perception is correct, why is this?

In a sense, these questions are a continuation of my thinking on authority and reason, and the minor point I made in response to Eduardo's post.