The debate focuses on what is reactionary, and what is not; and what is the essence of being reactionary - therefore, being able to answer the question: what (if anything) unites reactionaries?
The underlying assumption is that there is something coherent behind 'reaction'.
Yet the term suggests not.
What is reaction reacting against?
The answer is Leftism, Liberalism, Progressivism, Socialism - that general trend which could also include Protestantism (some aspects of), Deism, Atheism, Republicanism, Democracy and so on. (I prefer to call all this Leftism.)
So that which reaction is reacting-against is itself diverse.
And reaction itself is very diverse - depending upon what are the dominant aspects of Leftism its reaction focuses-upon.
So, the obvious answer is that reaction is unified only by that which it reacts against - reaction itself is unboundedly diverse, and coheres only in terms that all reaction point-towards that which it opposes.
So there is no essence to reaction - its cohesion lies outside of itself.
This - I submit - is why reaction is futile. The proper response to Leftism is not reaction, but that to which Leftism is itself a reaction: Religion.
The only true 'Right-winger' is not the reactionary - who is in essence merely anti-Leftist, merely aiming at the destruction of Leftism - but the Religious: who is prior to Leftism, does not depend on Leftism in any way, and is positive and constructive in stance.
(This analysis leaves open the question of 'which religion?'. There are at least several possibilities - but the religion of the Right must be prior to Leftism, and not a reaction to it.)