Hitler was the leader of the National SOCIALIST Workers Party (the German is abbreviated as "Nazi"). The difference between communism and socialism is that under the latter, the government directs production, while in the former the government both owns and directs the means of production.
Nazis in communists fought in the streets for control of Germany over the same pool of potential support - German socialists. Many of the Nazi party's early supporters were formerly communists. They did not defect from left to right, they defected from one form of socialism (International, i.e. Soviet-dominated socialism) to another (German national socialism). Hitler developed the latter to prevent the USSR from making a puppet out of Germany, giving "good Germans" a nationalist alternative to Soviet-dominated communism. Theirs was not a battle between "left" and "right," it was a battle between to peas in a pod, without enough room for them both.