They don't exist.

the dumbest in the smart row.

A liberal conservative is a conservative who leans toward the liberal side
and the other is a liberal who leans toward the conservative side

The first word is an adjective that modifies the second word

Similar to saying a liberal Republican
and a conservative Democrat

Words were meant to convey ideas. You can string them together any way you like as long as you get your point across.

Actually, I'm asking as to the ideologies of the two. Do you think there are any real differences?

I like what bluesbassman said in post #7

But I'll add this

IF you cann yourself a conservative liberal you still, are not calling yourself
a conservative.

Less confusing: If you are a conservative Democrat you are still not
calling yourself a Republican

Then there is always the differentiation that was made by Bluesbassman
which you could say either complicates things or is a different way of looking
at the distinction between the two.

But I do prefer "antimatter"!

lots of conservative liberals, though.