Following my negative definition of fascism as a secular (not religious), anti-egalitarian (reacting against Leftism, to at least that extent) and non-monarchial form of government...
(Insofar as fascism is monarchial, religious and/ or egalitarian - it is to that extent not fascism. of course, pure forms or types of political system are seldom/ never seen in reality - or at least not for long...)
, and considering the discussion in the comments,
there arises the matter of whether fascism was nationalist?
The answer, I think, is that successful fascism was nationalist.
Because fascism is a secular form of government, unless there is a strong nationalist sentiment, fascism cannot achieve the cohesion necessary to defeat Leftism.
And this is why fascism was a temporary phase during the early/ mid twentieth century - because strong nationalism is merely temporary (immediately post-religious) phase, and rapidly dwindles in just a few decades.
(Modern nationalism is merely a sub-type of Leftism - the self-award of victim status to a whole nation.)
So, a non-nationalist fascism was possible, but it would probably be small and weak. The fascisms which were successful enough to get themselves a piece of power were nationalist.