Yes, “United” should be in the name. In contrast, the Church of Christ is almost the polar opposite. They are a conservative evangelical denomination. I know many who belong to the Church of Christ. Some of my very best friends have been Church of Christ. I have theological issues with them because they tend to be legalistic and they add works to grace, but in terms of social/political matters, most are in lockstep with the typical FReeper.
That is good to know because I have friends in the Church of Christ and I’d hate to think they are in a messed up church!