earth_wizard's third principle of conservatism states that "the preference for liberty over equality is the most difficult part of the definition of conservative for most people to understand, particularly since liberty and equality are almost used as synonyms in our times. Put simply, all societies face a fundamental choice between emphasizing freedom or emphasizing equality."
I commented a little on this in my response in the earlier thread. Basically, I don't see liberty and equality as either synonymous or antonymous (or even hieronymous). Rather, I see them on different axes of what I shall probably conceive as a three-dimensional continuum of politics. I once tried to work out 3D continua for space, time, mass, energy, life and mind as part of the thinking around a song I once wrote. I got as far as time and part of mass and then broke down, partly because I was trying to make the continua mutually interdependent. Such are the things my mind does to me when I'm trying to vegetate peacefully. Woe et cetera.
Anyway...liberty occupies one end of one axis, and on the other I place security. I think this is a reasonable opposition: at one end you have a "society" where anyone can do what they like and therefore nobody is safe: on the other you have the ant-world of T H White where "everything not compulsory is forbidden" and vice versa, but where as long as the laws are adhered to nobody has anything to fear. Absolute chaos versus absolute order. Clearly the ideal is somewhere between the two extremes, and here I think conservatives occupy a sizeable spread, because there are some who incline more towards liberty and others who incline more towards law. (Also there are some things which some people think should be freer and some that some people think should be less free. Nothing is as simple as I am making it sound.)
So, on to equality, and here I am talking about equal status under the law, not any other kind. I will cover the other kind when I come to talk about exceptionalism. What would be the opposite pole of that axis? Well, presumably a society in which nobody was equal to anyone else, a hierarchy in which each level consisted of one person. So, absolute equality versus absolute hierarchy. Here I think I am well towards the equality end, if not actually at it. I certainly don't believe that being part of the government should confer any extra status on anyone, and here I differ from just about every society that to my knowledge there's ever been, including the allegedly "socialist" societies of Russia and China. We like to set our leaders above us, and exalt the person as well as the office, and the leaders themselves aren't going to be modest about it, especially when it means they can vote themselves salary increases.
I think conservatives are more prone to this hierarchical thinking than liberals, though, so yes, they do on the whole value liberty more than equality. But it isn't because they're mutually exclusive. On the contrary, I think that if everyone were equal under the law it might well lead to an increase in liberty.