Simply because two political ideologies are left or right wing does not mean they are aligned and therefore would protect each other.
Now, are there right wing political dictatorships? Yes there are, Pinochet of Chile in the 1970's is an example.
Fascist ideology focuses on elevating the State over all individual identity which is consistent with other left wing ideologies such as Marxism. Conversely, right wing ideologies elevate the identity of the individual above the State.
Now people will ask, "how are Pinochet and Hitler the same?" Both murdered political opponents, and both banned trade unions?" Again, you must examine the ideological motives of the action to determine if they are congruous. Pinochet banned trade unions, and privatized state owned companies. Hitler dissolved trade unions, but replaced them with the German Labour Front, which aligns with Fascist ideology in that there can be no more primary identity aside from the State. If you're a member of a union, say the steel workers union, this gives you an identity outside of the State, here fascism will reassert by destroying the union and reorganizing under a State banner.
I asked the following question a couple weeks ago to people who are adamant that Fascism is a right wing ideology, with not one person being able to respond: If Fascim is a right wing ideology, how do you explain the elevation of the State over the individual, when right wing ideologies elevate the individualism over the collective?
I suggest reading on Giovanni Gentile, he was the philosopher who created the fascist doctrine.