Spanish fascism was extremely Catholic (although the argument can be made that Francoism was not fascism.) Francoist symbolism was largely routed in Spain's strong Catholic background. This Catholic imagery and rhetoric was used to get a popular movement behind Franco, despite it being know that Franco was not extremely Catholic. Again, in this way and many others, it is important to draw a distinction between Francoism and the other far right European movements of the time. Spain, more than any other of the so-called fascist movements, sought to bring Spain back in time to idealized past (a key tenant of fascism). Nazi occultism was very ideological, and rarely came into play in the real world, while Francoist Catholicism played a very real, practical and serious role in the nature of Franco's regime.

The occult fascination with Nazism was mainly a postwar phenomenon, albeit with origins in the early 1940s. The link was created by the West's morbid fascination with Nazi war crimes and Hitler's cult of personality, in addition to the reuse of Druidic runes by some Nazi-associated thinkers. By and large, this link, and the associated mythology, is more sensational than factual, more late-night history channel than academic.

The same is likely true with Mussolini's version of fascism, which was more ecclesiastical than occult. He held rallies where the crowd would pray for blessings crying, "O Duce!, O Duce!, O Duce!". He also maintained a strong connection with the church, pandering to their desires for increased modesty in women's fashion and informing the pope of his desire to "control" Israel. This worked to such an extent that Pius XI deemed Italy's invasion of Abysinnia, a "holy war". So, to answer your question, Mussolini may have had fleeting connection to an occultist figure but his most important "spiritual" connection was with Catholicism and the Catholic Church, not the occult.

Source/further reading: The Pope and Mussolini: The Secret History of Pius XI and the Rise of Fascism in Europe by David Kertzer

Italian fascism? Oh gosh, yes!

Julius Evola jumps prominently to mind, although he was never a member of the Fascist Party (and was at times actively opposed to it, branding them "peasants", and criticising their violence and pacts with the Church). He was however a Far Right fellow traveller, flirting with the work of the Futurists (albeit briefly, his real love was Dada) and proto-fascist and revanchist figurehead Gabriele d'Annunzio.

During the 1920s, Evola became increasingly interested in mysticism, alchemy, yoga, tantra, Buddhism, Mithraic mystery cults, Greco-Egyptian magic, and more, releasing copious translations of older texts, as well as volumes that dealt with the spiritual transformation of the individual in a world of vulgar scientific certainty (a shot across the bows of the Futurists).

In 1927, Evola founded (with Arturo Reghini, who became increasingly reclusive into the 30s and 40s) the UR Society - UR corresponding to the characters of primordial creation - which gathered a dream team of mystics under its banner, and published a periodical entitled Pagan Imperialism. Although the UR Society was dissolved as a result of Evola's frequent potshots at Fascism and the church, Evola then threw his time into a journal called Krur (from the Sumerian for "strength") which had a more explicitly nationalist agenda, as described in one 1928 editorial:

Krur intends to link itself more explicitly to a vaster movement, which is affirmed on the one hand by the philosophical work of J Evola… and on the other hand acts in diverse instances to integrate into the preceding impulse of national renewal the values of spiritual, Ghibelline, heroic anti-European imperialism.

Even in his work on ritual magic (such as Introduction to Magic: Rituals and Practical Techniques for the Magus), there's a strong Nietzschean theme:

The figure of the magus retains in a highly visible way the ideal of spiritual virility, which is most essential for the higher type of the initiate or the adept. The magus has always called to mind the ideal of a dominating superiority.

Despite his criticisms, Evola was more supportive of the regime than not and his dream of dreams (particularly prior to the Lateran Accords being signed in 1929, which reconciled the Church and State) was that Italian Fascism would reject Christianity and embrace the Roman paganism of their proverbial forefathers.

Impressed by his criticisms of the Nazis' "scientific racism", Mussolini (who was tremendously well-read, and despite the trouble Evola caused held him in some regard) adopted Evola's more esoteric take on the matter (race wasn't just based on blood, but on the soul) and in 1942 commissioned him to produce the definitive Fascist racial philosophy, designed to bring Italian Fascism in line with National Socialism.

(As an aside, Evola considered Germany his spiritual home and maintained good relations with members of the SS. Himmler commissioned an investigation into him from SS-Oberfuhrer Karl Maria Wiligut, one of his pet occultists, who eventually concluded that “Evola works from a basic Aryan concept but is quite ignorant of prehistoric Germanic institutions and their meanings.” Choosing then, to reject his doctrine.)

Flushed by his recognition from the inner circle (he had been quite put out by his earlier rejections by the SS and the Fascists), he proposed a joint Italian-German journal called Blood and Spirit, which received Mussolini’s blessing.

He left for Berlin to begin work on his new venture, before his brief flirtation with mainstream acceptance came crashing down and he was recalled by a foreign ministry panicked by some of his speeches. Having had his fill of both Fascism and National Socialism, he left politics and began writing a book on Buddhism.

Sources: Magia Sexualis: Sex, Magic, and Liberation in Modern Western Esotericism by Hugh B. Urban, Fascism Viewed from the Right by Julius Evola and Against the Modern World: Traditionalism and the Secret Intellectual History of the 20th Century by Mark Sedgwick

As has been said already, Spanish Fascism, the "Falange" was completely outmanoevred by Franco. Franco was happy to let Antonio Primo de Rivera be executed, although he made enough of an attempt to rescue him to create an impression of trying to do so. He then ensured that the leadership of the Falange were entirely sidelined and that whatever revolutionary sentiments that Spanish fascism had were suppressed. Franco's regime was a conservative/catholic/military organisation, organised to maintain Franco in power, to maintain the army in a position of authority, to maintain the land and property owning classes in a position of authority, and to use the church to reinforce this. Paul Preston in " Franco: A biography." as well as in " Revolution and War in Spain, 1931-1939", "The Spanish Civil War: Reaction, Revolution and Revenge", and especially depressingly in "The Spanish Holocaust.". The closest I can think of Francoist ideology is that of a ruthless and murderous Daily Mail reader. It certainly wasn't fascist.

As it hasn't been mntioned yet: Austrofasicm was also very close to Catholicism while at the same time being fascinated by Mussolini's movement in Italy.