Going from socialism to fascism never was a long walk.
Ideologically, it's a pretty giant gulf. Mechanically, there are some socialist models that are similar (corporatism is the key similarity there, though not all socialist models embrace corporatism in practice, and fewer do by name.)
OTOH, fascism isn't a long mechanical walk from lots of modern government forms.
Of course, all government should always be suspect of overstepping it's boundaries, but I still think that countries with common law don't fall into centralized tyranny as easily.
Mussolini, arguably the inventor of modern fascism, began his political life as a socialist; to his last days, he maintained a belief in the role of the State to direct development and economic output.
In many ways, the economic policies of Fascism (and Nazism later) might well be described as "socialist policies run by - and for - oligarchic capitalists".
It would be more accurate to to describe it as oligarchic capitalist policies executed through corporatist institutions (which is very close to the non-Leninist socialist critique of Leninist-style Communism); the similarity to state-heavy forms of socialism isn't socialist policies, but corporatist institutions — that party, state, labor unions, and firms, and any other institutions permitted in society are expected, even if they have nominally separate character, to be coordinated and working together with a unified vision toward a unified goal.
After the split from Labour and failing to see success in his own New Party he went on a study tour of Italy and it's here you see the sharp turn towards the right wing and the formation of the BUP which was obviously authoritarian corporate fascism, strongly anti-Communist, and frequently clashed with Jewish groups.
As dragonwriter notes in his posts in this thread, corporatism is something that's sometimes shared among these ideologies but the gulf between those ideologies can still be pretty large, authoritarian nationalism inherently excludes certain groups in societies and is worlds apart from some other forms of socialism even if the goal of both is to end or reduce class conflict.
It looks like postwar Nazis didn't have much to consistently complain about regarding the retroactivity of Nuremberg trials:
> "Concerned with presenting the massacre as legally sanctioned, Hitler had the cabinet approve a measure on July 3 that declared, "The measures taken on June 30, July 1 and 2 to suppress treasonous assaults are legal as acts of self-defence by the State."[55] Reich Justice Minister Franz Gürtner, a conservative who had been Bavarian Justice Minister in the years of the Weimar Republic, demonstrated his loyalty to the new regime by drafting the statute, which added a legal veneer to the purge.[k] Signed into law by Hitler, Gürtner, and Minister of the Interior Wilhelm Frick, the "Law Regarding Measures of State Self-Defence" retroactively legalized the murders committed during the purge.[56] Germany's legal establishment further capitulated to the regime when the country's leading legal scholar, Carl Schmitt, wrote an article defending Hitler's July 13 speech. It was named "The Führer Upholds the Law."[57][58]"
Also at the end of WWII when Mussolini was deposed he became head of the Italian Social Republic[0], not named by accident.
The party that ended up as the parliamentary right in Italy was called Movimento Sociale Italiano[1] ("italian social movement).
One of the fascist mottos was “Tutto nello Stato, niente al di fuori dello Stato, nulla contro lo Stato.” (Everything in the State, nothing outside the State, nothing against the State).
This not not to say of course that fascism and socialism are the same, they are not, just that they shared a lot of background ideas.
[0] https://en.wikipedia.org/wiki/Italian_Social_Republic [1] https://en.wikipedia.org/wiki/Italian_Social_Movement