I suppose the part of the ends being "connected" is merely hypothetical, in regards to what would probably happen after.
Actually, no it is not hypothetical. My earlier post gave historical examples of one extreme giving way to the other. In practice, once you are at either extreme (Anarchy or Dictatorship) on the Spectrum you're referencing, the easiest path from there is always to wrap around to the other extreme.
In practice, a truly effective dictatorship is incredibly difficult to "rehabilitate". It happens, but dictatorships tend to be brought down by the mob or externally, also leading to mob rule in at least the short run. A VERY recent example would be Iraq, where Saddam Hussein ran a brutally effective dictatorship. Here effective refers to his ability to maintain control, and not his policies (which were inhumane, monstrous, and evil). Once the US toppled him, anarchy reigned for a fairly significant portion of time with the mobs exerting much more control over day to day life than anyone else.
I mentioned the Reign of Terror, but let's talk more about that. After Bastille day, the situation in France was essentially mob rule as the only existing form of government (Monarchy) was actively hunted down and killed. Eventually, the mob gives way to something else. In France, it was the Reign of Terror where folks were systematically and randomly accused and killed by committee, followed by Napolean, who was a classical military dictatorship.
The connection isn't just hypothetical.
I'd also say, you're a bit too quick to dismiss the two dimensional spectrum given here. You've purposely misunderstood how the x/y axis works. Can I explain further?
In the two dimensional version, the vertical Y axis represents the push pull between Libertarian-ism vs Authoritarianism. This means transition up and down vertically corresponds to government control.
Vertically, the Extreme Authoritarianism example is dictatorship. Without question. The extreme Libertarian stance is Mob rule.
The Horizontal X axis represents the push and pull between Cultural innovation (Liberalism) versus Cultural Traditionalism (Conservatism). Transition Left represents the willingness to innovate, sometimes just for innovation's sake, while transition Right represents how steeped in tradition the culture as a whole is.
Horizontally, the extreme Left is represented by the stereotypical vapid consumer. Think that tech geek that would say stuff like "iPhone? That's so 2008. I have an HTC Dream with Andriod Ice Cream Sandwich optimized..." In that case, it is innovation for innovation's sake resulting in a culture of disposable things (and often people). The extreme Right is the stereotypical Old Fart saying things like "In my day we went uphill both ways in the snow with no feet!" The past is romanticized and clung to with innovation being extremely distrusted.
How these interact is interesting. An Authoritarian/Conservative government would be an Authoritarian government deriving power from past tradition. A non-controversial example would be a dynastic monarchy. The right to rule absolutely is derived from past tradition, not embedded in even the qualities of the current genetic heir to power.
An example of a Liberal/Libertarian government would be a population that changes forms of government at a whim. Tossing aside one government for another as the Mob rule saw fit with no respect for past history or tradition. This is probably the closest you can get to a "true Democracy", which everyone fears as a form of government for a reason.
It is worth noting, that even here the extremes tend to meet. If you're familiar with the Reimann Mapping Theorem (you can map the 2 dimensional plane one to one onto a sphere) this makes sense. But the general idea is you can go off an edge and wrap back around to the other side when your system inevitably fails in the extreme.
