'Communism' and 'Fascism' are words which are thrown around by both sides. What the people who use them so carelessly seem to mean by them is 'bad', or 'authoritarian'.
Of course, it's a free country (still), so anyone can use any word as they like. But we would have clearer thinking if we used the word 'fascism' in a historically-accurate way.
Up until the First World War, the Right was for tradition, order, slow (if any) change. 'Throne and Altar' (Monarch and Church), plus the Army, was what the Right defended. The Left originated among those who wanted a republic, a constitution, free markets, an extended franchise, free speech.
It's a huge oversimplification, but we can say the Left at that time represented the new classes: merchants, factory owners, the professional middle classes ... as against the Right, based in early social forms: landowners, large and small.
But as capitalism developed and the peasantry started to be replaced by the industrial working class, a new form of 'Left' grew up: the socialist Left, which eventually displaced the original 'bourgeois' Left.
Up until the First World War, this Left was democratic: it championed civil liberties, a wider franchise, peaceful resolution of international disputes.
World War One changed all that. The Left abandoned its internationalist ideals -- each socialist party supported its own government. With one major exception: Russia, where one wing of the socialist movement -- the Bolsheviks -- opposed their own government, and in 1917, seized power in Russia. The chose the name 'Communist' to distinguish themselves from the socialists.
The world socialist movement split, with many socialists becoming Communists. In Russia, the ruling Communists ended up suppressing all other parties, and running a one-party state. (Why this happened is more complicated than it might appear, but that's another discussion.)
So now the Left had two divisions: 'democratic Socialists', and Communists. The former believed in the parliamentary road to socialism. The latter believed that a revolution would be necessary to achieve socialism.
And the Right split as well: seeing the growth of revolutionary elements who were internationalist -- ie anti-patriotic -- in their own country, some forces on the Right departed from the old pro-traditional structures Right, and embraced both the tactics of the Bolsheviks -- a 'combat party' -- and the general social program of the Left: a demagogic superficial anti-capitalism. Thus Mussolini (who had been editor of the pre-war Socialist Party newspaper), and Hitler, who formed the National Socialist German Workers Party.
That's really the origin of 'fascism': a new method of fighting the socialists, using the organizational techniques of the Communists and stealing the general social program of the Left as a whole, but with no intention of actually destroying capitalism.
The word 'fascism' was later broadened to include movements which had the same goal as the fascists -- a one-party state where the unions, the socialist and labor parties (and all others) would be outlawed -- even if these rightwing forces didn't embrace the 'leftist' social program of the original fascists: Franco's movement in Spain is an example.
The Communist and pro-Communist Left has never believed in free speech or civil liberties in general. But the non-Communist Left certainly did. Now that has changed.
We have a new 'Left' which is authoritarian, AND which despises its own working class. It's organized around 'identity politics' -- me-me-, and wants to dissolve all the traditional structures of society. (The old Communists and Socialists were strongly pro-family, and rather prudish in sexual matters. They also strongly embraced reason and science, whereas the new 'Left' is moving rapidly away from belief in objective reality and science. And the old Left was strongly pro-working class, even though the working class in their day was far more socially backward on racial and sexual issues than today's working class.)
So everything has changed. Using 'fascist' or even 'communist' to describe our new 'Left' actually confuses things. We're dealing with a new phenomenon. We don't yet have the words to describe it.
They oppose free speech and civil liberties in general. They have contempt for their own working class. In this sense, in a formal sense, they resemble the old fascist movement. But their origins are quite different, they don't have a political goal other than eating away at traditional society, they are anti-patriotic therefore ... so neither 'fascist' nor 'communist' really describes their essence.