It is a common misconception that the rebels/Nationalists led by General Franco were fascists, a misconception aided and abetted by Communist propaganda; the rebels' acceptance of aid from Fascist Italy and Nazi Germany did much to further this.
The only component of the Nationalist movement that actually subscribed to fascist ideology was the Falange (more properly, the Junta Ofensiva Nacional-Syndicalista, "National-Syndicalist Offensive Union"), whose program was essentially similar to that of the Italian Fascists under Mussolini. They were, however, a minority; aside from the Army, which was reactionary and Catholic where it was not Moroccan*, most of the combat troops in the Nationalist ranks came from the Requetes of the Carlist movement, who sought the restoration of the Carlist line of Bourbon pretenders, and thus were also politically reactionary and Catholic, but not necessarily in agreement with the generals. Add to this already volatile mixture the center-right Catholic bourgeoisie who felt pushed into rebellion by the Second Republic's anarchist and Communist factions, and you had a combination of factions that hated each other only slightly less than they hated the Loyalists/Republicans.
It is a testament to Franco's political skill that he was able to maneuver himself into leadership of what would soon be named the Falange Espanola y Tradicionalista y de los J.O.N.S. (Traditional Spanish Phalanx & The JONS) although the political cooperation was never as smooth as the military.** It didn't hurt the solidarity of the new Falange that the Republic made no effort to restrain the anticlerical excesses of its anarchist and Communist factions; these excesses alienated the still largely Catholic bourgeoisie and proletariat, and contributed to the myth of the "fifth column".
*About a third of the Army of Africa's strength was made up of 10,000 Moroccan regulares recruited in the African colonies; the remainder was composed of the Spanish Foreign Legion and Spanish troops stationed in Morocco.
**This was noted in a contemporary joke: before the unification of all Nationalist parties in the FET y los JONS, the Carlists usually wore a blue beret while the Falangists wore blue shirts. After the unification, all party workers were required to wear both beret and shirt. This usually resulted in hardcore Falangists stuffing the beret of the hated Carlists into their pockets when they could get away with it. As for the Carlists, they say that one old Requete was accosted by some young Falangist toughs, who shouted, "Hey, old man! Why are you out of uniform?" The old man turned to them and answered softly, "Because I cannot stuff my shirt into my pocket."