The Nationalists - i.e. Spanish Falange Fascists, as well as quasi-Fascists like Franco, had the support of conservative monarchists, the Church, and industrialists. They also had the support of Hitler and Mussolini - who provided arms and troops (especially fighter pilots).
The Republicans - Liberals, Socialists, Communists, Anarchists - had the support of labor unions and radical student groups, and had the backing of the Soviet Union.
So if Fascism is "Left Wing", what does that make the Spanish Civil War? A fight between different Leftist factions? And if so, why did groups that everyone acknowledges as Right Wing (monarchists, traditional Catholics) align with the Falange?
"conservative monarchists, the Church, and industrialists" supported one side or the other in the hope of exercising some control over it after it won. In general, that hope was often in vain ... although Franco was generally good for Spain. That's why communists in American media (Chevy Chase, I'm lookin' at YOU) mocked his death.
Yes, that's exactly what it was. Likewise Hitler and Stalin were allies at the beginning of WWII, each knowing they would eventually have to fight the other for control of Europe after France and England were defeated. Hitler stabbed Stalin in the back probably because he was convinced Stalin was about to stab him in the back. WWII was a war started by two socialist totalitarians, who had competing visions of a socialist world after they had defeated the Capitalists.
And if so, why did groups that everyone acknowledges as Right Wing (monarchists, traditional Catholics) align with the Falange?
Monarchists right wing? No, only when forced to be by the people. In fact the various French revolutions were short lived, because the monarchists were always undermining the newly formed French republics.
Catholics right wing? Not in the early 20th century. Catholic parishioners in the early 20th century politically did whatever their priest told them to do. The Catholic hierarchy propped up whatever government that served the church's interest, typically some German monarch. Traditional Catholics only moved to the right politically in the 1980s, when they realized that the Left despised Catholic morality.