My local UU "Church" is called "The First Religious Society - Unitarian". What the heck is that? It so happens that most of the congregation reads the Bible and we have a somewhat tradional service but it stops there.
Perhaps your church was the result of a merger with a Congregationalist church? Or maybe Quakers (Religious Society of Friends?) Some Quakers have moved away from the Society's original Christian background and have become more Unitarian (mostly the really liberal peacenik Quakers).
The Congregationalists (but certainly not the Quakers) might have bequeathed your church a more liturgical service . . .