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Merriam-webster describes reactionism as of, pertaining to, marked by, or favoring reaction, especially extreme conservatism or rightism in politics; opposing political or social change. And according to wikipedia, a reactionary may be described as:

In political science, a reactionary is a person or entity holding political views that favour a return to the status quo ante, the previous political state of society, which they believe possessed characteristics that are negatively absent from the contemporary status quo of a society.

The latter part of the Wikipedia definition reminds me of a slogan that was popular in the 2016 United States Presidential Election campaign of Donald Trump, Make America Great Again, but I am not aware of this campaign or its supporters self-describing as reactionary.

Rather, I seem to have encountered reactionary and reaction only as a description by opponents, in particular opponents on the far left, such as in the book titles Between Anarchist Rebellion and Fascist Reaction or The shipwrecked mind: on political reaction. I've also heard the term used by conservatives to describe others on "their side" of politics, in the sense of, "that opinion/group is not/no longer conservative, it has become reactionary", after a previously centrist or centre-right party or group has shifted toward the political right, or is perceived to have done such.

As a label, are there any political parties or groups that self-describe as reactionary? The Reactionary Party of Reactistan? The Movement for a Reaction? Reactionaries United?

It may sound nonsensical, but The "dark enlightenment movement" does.

Mencius Moldbug is widely credited with founding neoreaction (NRx), a political philosophy and intellectual movement dedicated to providing secure, responsible, and effective government. A central thesis of neoreaction is that accomplishing this goal requires a critical re-evaluation of “democracy” from the perspective of political engineering and informed by a study of the great political thinkers of the past.

As Moldbug puts it in A Gentle Introduction to Unqualified Reservations :

The essence of any 21st-century reaction is the unity of these two forces: the modern engineering mentality, and the great historical legacy of antique, classical and Victorian pre-democratic thought. The adept, to achieve reactionary enlightenment, observes that both yield the same result. What can it be, but the truth for which all good men seek? Armed with this sure and fearless faith, the Reaction conquers all.

The above is a self-description on his homepage.

Another example would be:

The Impossibility of Conservatism; or, Why I am a Reactionary

Perhaps a bit surprising:

Khomeini on reactionary political thought:

Yes, we are reactionaries, and you are enlightened intellectuals: You intellectuals do not want us to go back 1400 years. You, who want freedom,
freedom for everything, the freedom of parties, you who want all the
freedoms, you intellectuals: freedom that will corrupt our youth,
freedom that will pave the way for the oppressor, freedom that will drag
our nation to the bottom.
_{EconLib: "3 Fun Quotes from Ayatollah Khomeini", 2015.}