Foreword (skip if you like)
First I have to comment that I do not like the horseshoe theory. If I look for instance at the corresponding Wikipedia article, you find as examples of modern use the prevalence of anti-Semitism on the extreme right and left, as well as a citation of Josef Joffe (issuer of the largest social-liberal German weekly newspaper), who claims that the German left party "Die Linke" has similar political positions than the Austrian right-wing so-called 'populist' "FPÖ".
These claims are contradicted by empirical findings: while there is typically a rising sympathy to the Palestinian cause on the Left, usually the proportion of people that hold anti-Semitic preconceptions typically declines when going from right to left (at least in European countries). This of course highly depends on how you define left and right, but the anti-correlation between upholding progressive values and anti-Semitic preconceptions is pretty well established.
Also the second claim is clearly contradicted by evidence of just comparing policy recommendations of the parties; the Linke has almost no policies in common with the FPÖ or other right-wing 'populist' parties in Europe such as the German AfD. However, it is rather the conservative parties that share quite some policies with the the right-wing parties in the fields of immigration, inner security and policies concerning civil liberties.
Maybe this theory has some merits in explaining some political phenomena, but I really do not seem how this oversimplification helps much in understanding anything. To me it seems the real merit of this theory is for the ruling class and those sympathetic to them to discredit both, the left and the right, at the same time, killing two birds with one stone.
Suggested clarification of terms
As for an economic version of the horseshoe theory, I first have to clarify what the term "Communist society" means. Apparently you don't use it in the sense Communists and Marxists use it, because then the question would be absurd. A Communist society, in their definition, is a society where social classes, money, and the state are absent. The means of production should be owned by the workers themselves. Obviously, there never existed a Communist society that fits this definition that we are aware of.
So you are probably referring to the societies that existed in the former Sovjet Union or the Eastern block states, which were governed by their Communist Parties - which doesn't make them Communist states. The Communist parties tried to achieve a Communist society, but departed from that route during their Leninist and Stalinist doctrines.
The economic system of these societies is characterized by state ownership of the means of production, and centralized administrative planning, societies which were called by Marxists usually as "Actually Existing Socialism"
Marxist critics of the Leninist and Stalinist route, called Trotskyist, usually refer to the societies that fall under the term "Actually Existing Socialism" as state capitalist or state monopoly capitalist societies. The development of the Sovjet Union and related states never reached beyond such a state monopoly capital phase.
The neo-Trotskyist theory
The theory that probably gets closest to your economic version of the horseshoe theory is the neo-Trotskyist theory:
The state in Soviet-type societies was redefined by the neo-Trotskyists as being also state-monopoly capitalist. There was no difference between the West and the East in this regard. [emphasis in original].
While there is some merit to this theory, you already see the that there is some huge oversimplification which does not hold up to reality.
Some merits of such a theory
In private companies, you have a fully totalitarian structure (regarding your "life" in the company, not your actual life of course. When you're fired, you are "dead" for the company). The major decisions in large corporations are made by the board of directors, a process that is completely in equivalence to the decision of the Politburo of the Communist party.
Within a corporation, all economic decision are centrally top-down planned just as in a state-planned economy. There is usually no inner free market in a corporation, where the divisions compete with each other in the same field. The competition between business divisions is more like the competition between different branches of the economy in a state-planned economy for a higher share of the budget.
Vertical mergers of corporations by buying up their direct suppliers or vendors make this part of the economy more look like a corresponding section of a state-planned economy.
1) The totalitarian corporation? - Review by Kean Birch. An excerpt:
The conceptualisation of the corporation as a ‘totalitarian’ institution is a more recent phenomenon, present in the work of authors like Joel Bakan, Ted Nace and David Harvey reviewed here. All three authors have, in one way or another, something to add to the view that the corporation is totalitarian, as well as illustrating how the processes and practices leading to our present situation are considerably more complex than any simplistic representation of corporate malfeasance would imply (i.e. that criminal corporations are individual ‘bad apples’ and not representational of the whole system)