I broadly agree with you regarding the psychology behind far-X turning into anti-Y at the expense of things that X supposedly believes in (where X,Y are left,right or right,left).

I'm much less convinced, however, by the justification for trueness vs. fakeness of right and left. You state that the key defining characteristic of the left is to be more inclusive, but provide no justification. The terms arose from the seating in the French legislature, where almost immediately the Girondin--despite being anti-monarchy--were purged at the (violent) behest of supporters of the farther left (in terms of literal seating arrangement). As an origin story for the terms, it is profoundly unsatisfying, and also not very aligned with your stated definition. The left precipitated the Reign of Terror, which hardly supports your claim that the hostility of the far left disqualifies it from even being "left".

Furthermore, in the U.S., it was the right, not the left, that advocated for the abolition of slavery--arguing for it largely from traditional religious values, while the left was busy with post-hoc rationalizations.

I agree that in the U.S., in the post-Civil-Rights-Act, post-Vietnam era, the left has for most of the time quite deeply embraced liberal ideals (hence the term "liberal" being an apt synonym), but I don't see how that's necessarily the true left. It is a flavor of left: the liberal left, one might say.

The far left is assuredly not liberal when it organizes itself around anti-Right views. But is it really not even left? If we're to say that--and we could, I suppose--we have also to say that fascists aren't on the right, because the degree of state control necessary for fascism is profoundly at odds with the anti-governmental attitude that has characterized the right over the same period of time for which the left's dominant ideology has matched your characterization.

We could do that, I suppose. I've even seen people argue that Hitler was a leftist because of this (much as you argue that Stalin is on the right). But isn't this just doing the anti- thing too hard? Just because someone on one side adopts some positions that are historically anathema to that side doesn't mean that it's on the other side. It can be its own bad thing.

So I think you dismiss the horseshoe a little too easily. If you take the common division of left/right into authoritarian vs individualistic and reform vs traditional, it's apparent that whenever you want to be so reform or so traditional that you'll get a lot of opposition, resorting to authoritarianism to force your way (the good way, the right way, the moral way, the superior way) becomes increasingly tempting. It's not like we never force our way--we tend not to tolerate murderers, for instance--so the problem isn't that it's never right to force your way ("don't murder people"), but rather the scope of what can legitimately be forced.

I prefer the terminology used by the Economist, which dubs the far left that you describe as the "illiberal left". This seems perfectly apt to me in that it recognizes both the heritage and the multiple nominal policy positions that resemble the liberal left, but also the departure from acceptance and open-mindedness into hostility and virulent opposition to "the right".

Addendum: Giles does provide somewhat more support for his view of Left and Right in an article posted after this one (https://medium.com/@dgilesphilosopher/new-book-what-are-left-wing-and-right-wing-812fe4c2d2d2); it does not actually address my core critique, but at least it gives a more in-depth explanation for what he views as the key point of contention between "Left" and "Right".