Related question: How accurate is the idea that the Nazi's were heavily involved in the occult?
I mean, we see a lot of portrayals of Nazi's as dark warlocks in a lot of popular culture, but that doesn't mean that occult practices or beliefs were actually that widespread in Nazi Germany. Is this popular depiction just propaganda meant to create additional contrast?
Italy
Julius Evola was on one hand one of the leading philosophers of Europe's neofascist movement in the mid-20th century, and on the other hand his self-styled "magical idealism" philosophy harbored a fairly eclectic mixture of beliefs in alchemy, supernatural, telepathy, etc.
That's not to say he was naive or uneducated. Some of his writings on Buddhism are acknowledged as the work of an authentic scholar in this field, although authors such as Arthur Versluis have criticised Evola for being biased, or for using this medium as a platform for his own agenda.
He also wrote extensively on topics such as Tibetan tantric yoga, the myth of the Holy Grail, Western esotericism, etc.
He was admired by Mussolini and Otto Rahn (SS officer and seeker of the Holy Grail - Indiana Jones and the Last Crusade alludes to Rahn via one of the secondary characters). In the novel Foucault's Pendulum by Umberto Eco there are some echoes of Evola in the depiction of Agliè, but without the connection to the far right (the novel is concerned with the esoteric circles in Italy post-WW2, up to the '80s).
The Eastern Bloc
There is no equivalent to classic occultism or hermeticism in the communist Eastern Bloc, since the official doctrine was marxist materialism and ideas not aligned with it were suppressed. However, a range of beliefs in "bioenergy", energy healing, remote viewing, telepathy, etc had a semi-official status, often focused around charismatic individuals such as Eugenia "Juna" (Djuna) Davitashvili, Baba Vanga, Anatoly Kashpirovsky, etc. In some (perhaps isolated?) cases, there was even (semi-open) support for these beliefs and these individuals from members of the scientific establishment in the East. There was a tendency to present this whole field as existing at the far edge of science - perhaps (at least in some cases if not all) as an attempt to placate censorship.
Djuna Davitashvili, who lived in the Soviet Union, had claimed to possess remote viewing abilities. It was widely rumored she was asked to heal Leonid Brezhnev and Boris Yeltsin using her "bioenergy" (what would be called "supernatural powers" in the West). Please understand that the very nature of this topic, combined with the secretive culture of the former communist countries, makes it very hard to say for sure whether these attempted "healing" sessions actually did happen (to say nothing of the actual efficacy of the "treatments").
Baba Vanga, a blind faith healer from Bulgaria, was more akin to a religious figure, a prophetess. It was claimed she could predict future events. She was the subject of a study by two Bulgarian psychologists, working for the government, attempting to figure out the veracity (if at all) of the claims surrounding her. There's no peer-reviewed science here, of course.
Anatoly Kashpirovsky was a "psychic healer" and hypnotist from Russia in the late 80s / early 90s. He was famous for his "mass healing" sessions, sometimes performed live on national TV.
Whether these individuals were an integral part of their respective cultures at the time, or whether they were more like isolated phenomena, is hard to say. The fact is they did exist and each one of them had some degree of fame or influence.
Does Italian fascism necessarily require esoteric beliefs a la Evola? Does the culture surrounding the Eastern Bloc political doctrine necessarily imply a belief in "bioenergy" or remote viewing? These are probably not questions for the field of history - seems more like cultural anthropology to me.
Follow up: same question for Japan and China