As someone from the world of political science I can maybe give you some insight into this. I’m new here but I’ll try to explain the best I can in a very general overview. This is an excellent question.

To start we can look at what 20th century leftist political movements were concerned with. They were not all communist. You had many, many different variations of socialism, communism, anarchism, different adaptations of Marxist theory etc.

These factions had often mutually exclusive world views and would even come into violent conflict with each other. The infighting among the left in the Spanish Civil War and Russia Revolution are good examples of this. That being said, today we often lump all these together as “communism” so I understand where you are coming from. I’m going to say “leftist” to keep it simple.

A defining feature of categorically left wing movements though is the distain for hierarchies. Specifically traditional hierarchies and the hierarchies they saw in capitalism. You know this as the idea workers should unite and run the factories (means of production) themselves rather than the capitalist owners who control them. But this goes beyond ideas about capitalism.

In countries that had leftist revolutions the political hierarchy was also of concern. This not only included their monarchy but also their respective religious institutions. In the Russian Empire the Orthodox Church was extremely powerful and played a major role in the power structure at the time. Leftists saw the church as part of the unjust hierarchy that had to go.

Granted many important leftist intellectual leaders were atheist or at least not a practicing member of any religion. Lenin was a good example of this. People who had studied the work of 19th and 18th century philosophers and theorists who saw religion as irrational or simply false were pretty common in that crowd. In particular in Paris where people like Lenin and Trotsky spent a lot of time (usually for reasons of safety or exile). Someone who isn’t religious or thinks religion is just false is a lot more likely to be willing to see the church as part of the unjust hierarchy and therefore must go. This irreligious mindset wasn’t unique to the far left but was very common.

Leftists also came from many walks of life including different religious and ethnic backgrounds. So they saw having a specific church in the loop in power as oppressive and unfair. Just as we today have freedom of religion a lot of these people were fighting for freedom from a specific religion’s political authority.

This is what explains the anti-clerical aspect of many leftist revolutions. It’s not necessarily that religion itself is the problem but rather that the clergy was an accessory to the crimes of the unjust hierarchy they thought had to go. So when leftist revolutions succeeded, even if only briefly, one of the first things they did was kick out the clergy from power along with the monarchy and capitalists. This is where we get the anti-religion stereotype from in the West of leftist regimes.

Now is your question off base? No, not entirely but perhaps misinformed from stereotypes.

For example in 1941 Joseph Stalin made policies to revive the Orthodox Church and in 1943 they began having elections for church patriarchs again. The church wasn’t in the same position as before the revolution but it was moved back to being a somewhat visible part of Russian and Eastern European power. Persecution of Jews in Russia also returned which was an unfortunate but common practice in the mostly Orthodox parts of Russia and Ukraine going back centuries. You can still see this today in Ukraine and Russia. Now you know why senior clergy are seen on TV with Putin every now and again. Today there are many Russian Orthodox Church goers that will bring ornate icons (religious paintings) of Stalin with them to services and many churches keep them inside as well. You can also see the Orthodox practice of having Stalin put in a glass case, well preserved during his long state funeral. This looks like odd, overly reverent leader worship to us but this is how a saint or important religious figure was treated traditionally.

Religious belief was still common among your average person who wasn’t a political intellectual. So it never really disappeared entirely and nor did the persecution between different religious groups sadly. Secularism is still less of a thing in the Eastern politics than in the West despite very active attempts to bring it about there in the 20th century.

Why didn’t they put a religious face on it like the NAZI’s (fascists)? That is a whole other can of worms but I’ll be brief.

The fascist movements in the 20th century didn’t need to put on a mask because they didn’t have this issue with traditional religious hierarchy in the same way. In reading the Doctrine of Fascism by Mussolini and Gentile you will understand why. Fascists for the most part saw religion as a VERY important part of the society. Religion was part of their identity as a nation and saw leftists and liberals as eroding it or perverting it. Now they didn’t want to share much power with the regional churches (mostly Catholic except in Germany where there was also a major Protestant population) but they didn’t believe the church was a bad thing. Fascists were explicitly anti-secular and often saw themselves as a movement to push against the rise of liberalism and leftist political factions. The Ustaše in Croatia were very open about their Catholicism along with the Slovakian, Italian and Spanish fascists. This didn’t mean they never came to blows with the church over certain power dynamics but they didn’t see it as an enemy either.

Unlike leftists the principle of the far right in political theory (in general) is that political, moral and religious hierarchies are just and necessary. So you can see why it’s different. They are genuinely the political opposite of the communists.

So to sum it up in general terms

• Not all leftists were atheists but were against religious institutions having any power due to their beliefs about hierarchy and their ethnic and religious diversity as a movement.

• Religion never really disappeared in the 20th century leftist regimes. It was even brought back in some cases such as with Stalin. It also remains today after the end or reforms of those regimes decades later.

• Fascists had an opposite view about hierarchies and religion so they didn’t have to fake anything about it. They just actually were religious and that was an important part of their program.

I know this is a more Eurocentric view of this but hopefully a political science perspective on 20th century history there helps.