A good monograph to pick up if you are interested in the intellectual origins of fascism is Roger Eatwell's Fascism: A History. Much of this answer draws heavily from that text.

Whilst fascism certainly has an anti-communist agenda, fascism's origins should be viewed in a similar manner to Marx or liberalism, that being the onset of European modernity. As Eatwell writes:

fascism was a negation of the Enlightenment, part of a counterrevolution that rejected the basic assumptions of "modernity"

Some of the Enlightenment principles that fascism rejects are quite identifiable: individualism, the constitutional state, laissez-fare capitalism, etc. But if fascism's roots are in the Enlightenment, where should we begin to look?

Many enlightenment thinkers have been singled out (both fairly and unfairly) in trying to pinpoint the intellectual origins of fascism. Some have pointed to Rousseau's belief that a "general will" in society could emerge. Others point to Hegel. Even more simply point to the historical discipline as a whole of planting the seeds for nationalism and imperialism that both fascist Italy and Nazi Germany would so readily cultivate. Whatever the case may be, a lot of historiographical effort has been made to view fascism as a set of ideas, and tracing the origin of these ideas back into the 19th century.

Now, when contextualizing the rise of fascism after the First World War, the role of communists and communism must be mentioned. However, in terms of the history of ideas, fascism is more complicated than merely a reaction to communism. It was a competing understanding of how to confront modernity, and its roots lay in the rise of nationalism and various intellectual traditions that predate the Russian Revolution.