Nailed it.Fascism is counter-revolutionary, in the sense that Joseph de Maistre defined counter-revolutions as "not ... a reverse revolution, but the reverse of the Revolution" - and as such is explicitly right-wing. It is the assertion of a particular hierarchy, based on national identity, in the context of that hierarchy's absence. That absence is universally blamed on forces and institutions of left-wing social corruption; Marxists, foreigners, Jews, etc. Only when those forces are overthrown and defeated can that hierarchy assert itself once again, and democratic politics is conceptualized as itself a corruptive force that stands in opposition to fascist goals. Thus fascism is autocratic in means.
It is nationalist in that it situates the protection of the national identity from these left-wing forces as the primary obligation of state power. It is socialist in that it works to mobilize the body politic into a single cohesive unit for the sole purpose of achieving this end. It is not conservative, in that it rejects orthodox traditionalism in favor of revolutionary traditionalism, but it is still very much right-wing.