Hitler and Mussolini were socialists, who abhorred communism.
Well, actually, you’re right that they abhorred communism but, of course, they weren’t socialists.
Socialism involves common ownership of the means of production, distribution and exchange and neither Fascism nor National Socialism were about common ownership of the means of production distribution and exchange, quite the opposite.
The German word for the English words ‘National Socialism’ is ‘Nationalsozialismus’, it’s a typical German compound word and there’s something of a hint in that. The Party platform was about returning Germany to earlier times, a world of peasant proprietor farmers, artisan/craftsman production and small shopkeepers (the contradiction was that they needed Big Business - which was a threat to peasant proprietor farmers, artisan/craftsman producers and small shopkeepers - to achieve that) and, indeed, the bulk of their support came from these kinds of people, the money came from Big Business because the Party promised to end the threat of Socialism, Communism, Trades Unions and so on.
A couple of German words for you - ‘Gleichschaltung’ (coordination) and ‘Volksgemeinschaft’ (people’s community) which express where the ‘social’ comes in. The aim was to re-create a (mythical) German community of the past where all the peasant proprietor farmers, artisan/craftsman producers and small shopkeepers would be unified, with one set of ideas/ideals - lots of independent people all thinking the same way with an end to divisive concepts such as ‘class’ and religious feuding. Anything social, political or racial that stood in the way of that would be eliminated and, since the idea was that Germans needed more room to have lots more peasant proprietor farmers, artisan/craftsman producers and small shopkeepers, it would stretch over Eastern Europe and much of Western Russia, the inhabitants of which would be eliminated or provide a slave class.
So, no it wasn’t about common ownership of the means of production, distribution or exchange, the peasant proprietor farmers would own their farms, the artisan/craftsman producers would own their workshops and the small shopkeepers would own their shops. They’d just all think the same way.
Italian Fascism was somewhat different but you get the general idea.
Were there some Nazis who had ‘socialist’ ideas and some Fascists who had ‘socialist’ ideas? Yes there were but there are Catholics who support he idea of women priests . . . . .