This does remind me a bit of the "infinite horsehoe" idea put forward by the Academic agent.
It was his attempt to explain why the "other sides" seem to horseshoe to each other from the perspective of a third side.
His three sides were summarized as Liberal, Socialist, and Fascist for the initial explanation, but when he breaks down what traits he's applying to the three sides of the triangle, its obvious he's discussing something a little bit broader.
Or at least, as someone mentioned earlier, Socialism and Liberalism are relatively clearly defined, but the final point of the triangle is much more catch alli. That pops up in several of his explanations.
For example, Liberalism supports negative rights, Socialism supports Positive rights. That's a clear enough distinction. However, the other side of that doesn't really have a clear, universal position on that.
I think maybe the main divider of "the third point" which separates it from liberalism and Socialism is a general rejection of Universalism: liberalism and Socialism both have a belief in a universal rule and order, and the argument is over what that universal order is (private property or public property, for example). The third way is much less accepting of that: for example feudalism puts a lot of things in muddled middle ground between private property and public property. And Monarchists implicitly reject a universal rule code for people, with explicitly different laws for different people.
Now, that might suggest were taking something that is a spectrum, and trying to make "centrism/moderation" a different axis, but saying "monarchy is a middle ground compromise between liberalism and Socialism" doesn't really sound totally right either.
