This previous answer I gave about Spain's place in the Cold War might interest you. The TL;DR is that Franco was a very smart man, whatever else you might think of him, and able to realign himself following the war to move into the American camp.

There is a lot to answer there, and I can only answer a small part of it, specifically in relation to the relations of Spain with the Soviet Union, or rather, the near total lack of them(!), and the United States as well (BFFS!). Franco's Spain was viciously anti-Communist. Part of the reason for the Civil War had been the belief by elements in Spain that the government was going to far left.

The Soviet Union had been - aside from Mexico - the only significant power to openly support and supply the Republican Forces during the Spanish Civil War. The other fascist powers - Germany and Italy - had supplied the Nationalist Forces, and while Franco didn't join in for WWII, he allowed for the raising of a volunteer force, which only was to fight on the Eastern Front, as it was portrayed as part of the crusade against Communism specifically, not a broader endorsement of the Axis' war aims against the Allies. Known as the Blue Division (Division Azul), the fought as part of the Heer from the very early stages of Barbarossa until late-1943, when they were recalled. It numbered about 20,000 men at its peak, and was generally very well respected by the Germans.

As I said though, Franco was doing his best not to offend the Western Allies, and they pressured him to recall the division. Internal pressure began to mount too, as the fight against the Communists wasn't going as well as hoped. The Blue Division was recalled in late-43, but many wanted to continue fighting, and were mostly integrated into the SS, as a foreign raised company, although others were distributed around the Heer.

In the post-war era, service in the Blue Division was still seen as a very positive thing. Although the government didn't explicitly endorse it - wanting to not remind the West of the episode perhaps - veterans groups held reunions and openly would assemble wearing their uniforms and decorations, and there are plenty of memorials to the fallen in Spain.. The veterans remained proud of their efforts to fight against communism, as anti-communist sentiment was still a very strong part of Francoist Spain. At least through the 60s, being a communist was a good way to end up in prison (The White Terror), and I believe that the Soviet Union didn't even recognize the government of Spain until legitimate until the 60s, and a quick search shows they only reestablished an embassy in the late 70s.

As for the United States, during the Civil War the US was neutral. Its well known that Americans joined the International Brigades of the Republican side, but it was illegal for them to do so, and they faced prosecution, or at the least blacklisting, at home in some cases when they returned. American sentiment was somewhat pro-Loyalists, but at least some American business interests threw their support to the Nationalists. In theory, I believe that support should have been illegal too, but I don't know if the companies (GM for instance!) ever faced punishment for it.

During WWII, the US wasn't exactly buddy-buddy with Spain, but their participation on the Eastern Front didn't really influence US policy from what I have read. If anything, the US seemed not to care all that much. Spain's support of Germany saw them ostracized for a short time in the international community, but the Cold War mentality soon took hold, and quite soon after the war, the US and Spain began to form some level of alliance in solidarity to their mutual anti-Communist leanings. While not the same level as NATO, the US was nevertheless sending them military aid beginning in the early 1950s, and the US was allowed to maintain military bases there, all thanks to the Madrid Pact, which was signed in 1953. They also pressed for Spain's admission into the UN, which the USSR had attempted to prevent, at least party in retaliation for the Blue Division, and more generally, because of the strong anti-Communism spouted by Spain.

So that is the foreign relations in a nut shell. Francoist Spain really hated Communism, and the USSR returned the feelings. The US on the other hand, with what some would say was a typical mentality for the era, didn't care much what kind of domestic policies were in place as long as they were an ally against world Communism. As for just how fascist Spain remained in the post-WWII era, I can't really offer much aside from, as I mentioned, the continued repression of Communism, so I'll leave talk about domestic policies to someone else.

Hope that helps. Be glad to expand on what parts I can.

I always tend to think that describing Franco as a fascist is at least an over-simplification, if not an outright canard.

Defining 'fascism' to any certainty is a perennial problem in political theory, and it's certainly true that in Francoist Spain you can see some of the hallmarks of what we understand that term to connote: a centralised single-party, authoritarian state rooted in nationalism and led by a strong leader and pursuing a corporatist, syndicalist economic policy.

But Franco's programme of national rejuvenation wasn't expansionist or militaristic, and it wasn't predicated on the same pseudo-scientific idea of racial supremacy as Italian Fascism or Nazism. Those ideologies were fundamentally forward-looking; Franco was a deeply conservative, devoutly Catholic character who was ultimately set on a restoration of Spain to a past (and probably mythical) ideal, not a revolutionary remaking of Spanish society — he conceptualised the civil war as a crusade, or a second Reconquista.

It's true that Franco's coalition, at least during the civil war, included fascist elements — just as it included monarchists, Carlists and mainstream democratic conservatives. But the Falange was quickly subsumed (and its leaders sidelined) into a larger Francoist movement which adopted, adapted and reoriented its ideological doctrines.

The serious historiography on Spain is reticent about calling Franco a fascist. Someone else in this thread has already referenced Paul Preston — I've read a lot of his work, and I really don't rate him as a historian. He's a hyperbolic writer who heroises Republican Spain while casting Nationalist Spain as a force of pure evil, without much in the way of nuance — The Spanish Holocaust, the basic premise of which is that nationalist repression during and immediately after the civil war constituted genocide, is particularly flawed in that respect.

This is Stanley Payne, one of the foremost Hispanists of the 20th century, on the question of Franco's fascism:

This does not mean that Franco was ever a generic fascist sensu strictu. More than twenty years after his death, Franco has still eluded precise definition save in the vague and general categories of 'dictator' and 'authoritarian'. Thus scarcely any of the serious historians and analysts of Franco consider the Generalissimo to have been a core fascist. Paul Preston, never known to give Franco the benefit of the doubt, once observed at a scholarly conference in Madrid, "Franco no era un fascista sino algo mucho peor" (Franco was not a fascist but something much worse).

Source: Payne, Fascism in Spain, 1923-1977 (1999), 476.

It's also worth noting that Franco's relationship with the Fascist powers was far from cordial. They poured money, arms and personnel into Spain during the civil war, yes, but they privately regarded Franco with some disdain.

... so, having said all of that, to answer your actual question (albeit slightly reframed): yes, Franco's ardent anticommunism made it a useful European ally for the US, just as the US allied itself with South American strongmen of similarly authoritarian types during the Cold War. It's worth noting, however, that the western European nations were far less accommodating, and Spain remained something of a pariah state within Europe until the end of the Franco regime.

A couple of extracts from a US government country study on the foreign policy of Francoist Spain, published in 1988:

As the United States became increasingly concerned with the Soviet threat following the fall of Czechoslovakia, the Berlin blockade in 1948, and the outbreak of the Korean War in 1950, United States policy makers also began to recognize the strategic importance of the Iberian Peninsula; furthermore, they realized that ostracism had failed and that the Franco regime was stronger than ever.

[...]

Spain's European neighbors were less willing than the United States to modify their aversion to Franco's authoritarian rule. The West European members of the North Atlantic Treaty Organization (NATO) vetoed efforts to include Spain. Spain's applications for association with the European Community (EC) were also repeatedly rejected. Although a Trade Preference Treaty between Spain and the EC signed in 1970 seemed to herald a thaw in relations, Spain's entry into the EC, continued to be a political issue throughout Franco's lifetime. Spanish membership in the Community, considered by Spanish economists and businessmen as crucial for Spain's economic development, had to await the democratization of the regime.

And lastly, on the transition to democracy: the decline of Francoism after his death was the result of a confluence of factors. By the 1970s, Spain had seen dramatic economic growth (especially during the '50s and '60s) but that development was stunted by its isolated position within Europe. There was a significant reformist faction within the FET y de las JONS. Those pro-liberalisation forces gained significant traction when ETA assassinated prime minister Luis Carrero Blanco, a hardline Francoist and the functional heir-apparent to Franco, in 1973. When Franco died in 1975, Juan Carlos essentially aligned himself with the reformists — it's not entirely clear that stemmed from some personal conviction, or simply because the momentum of change was with the liberalisers.

I'm surprised no one has challenged this outright yet, since neither Franco nor Francoist Spain were fascist. It is true that it relied on the support of fascists (particularly during the Civil War) and true also that Franco extensively employed fascist imagery and even occasionally tried to convince people of his fascist credentials, but he neither he nor his state were fascist.

My real area of experience is the Spanish Civil War during which Franco actually managed to neuter the fascist party, the 'Falange', and ensure his personal rule more than that of an ideology.

Before the war, military conspirators (namely General Mola) relied on fascist or fascist-lite parties like the Falange and CEDA to aggravate and attack leftists in order to promote disorder and chaos and justify their planned military coup. They wanted the opportunity to end the Second Spanish Republic which they saw as weak and, more personally, which threatened to reverse some of the promotions gained by senior officers during their colonial campaign to pacify Spanish Morocco. There was very little fascist drive behind this from military generals, though, and in fact the leader they planned to install was José Sanjurjo who - to put it no stronger - had monarchist sympathies.

But so what? Even if it wasn't planned as a fascist dictatorship, surely it could have become one over the course of the war? Franco, after all, relied on fascist arms, didn't he? Well, If nothing else, the war provided an excellent opportunity for Franco. Sanjurjo died in a plane crash, and so too did General Mola over the course of the war. Most fortuitous was the Republican capture of the leader of the Falange, though: José Antonio Primo de Rivera was his only rival capable of marshaling a real challenge. He was executed for his part in the plot but not before Franco had capitalized on the chance to discredit him. He put it about that the fascist leader had probably died cowering and apparently told one of Rivera's grieving aides that he had been castrated by the Russians.

After that, it was simple enough to restrain the fascists in the Falange: Franco supported a particularly naive candidate for leadership and arrested his rival. Thereafter he made it seem as if the new leader had handed Franco power willingly and concentrated the power of the party beneath him. Moreover, he began work on incorporating the Carlists (a particularly popular strain of monarchism) into the Falange and further diluting the fascism. The ridiculously named Falange Española Tradicionalista y de las Juntas de Ofensiva Nacional Sindicalista (FET de las JONS) was born, tying together the disparate elements of 'the right' and creating the single party of Spain until the end of the dictatorship.

If I had more time, I would point to Franco's poor relations with the fascist powers but instead I'll recommend two books that I've drawn from: Paul Preston's "The Spanish Civil War: Reaction, Revolution, and Revenge (Revised and Expanded Edition)" and Filipe Ribeiro De Meneses' "Franco and the Spanish Civil War."