Correctamundo.
The key ideological driver in both Communism and Nazism is the welfare of the People -- Volk in German, Narod in Russian -- narodny this, narodny that got stamped all over everything in Russia after the Bolsheviks took over.
That's the main discriminator between the Leftist Nazis and Communists, on the one hand, and Italian and Spanish Fascists and Phalangists on the other, for whom the preservation and empowerment of the State represented the highest value instead.
Germans and Italians collaborated during the war and even convened "fascist courts" to try traitors to the cause like Count Ciano, after Italy left the war, arrested Mussolini, and was promptly occupied by the Germans. But Germans and Italians alike always understood and were aware of the theoretical differences between Italian fascism and German Nazism.
T Sowell points out that Naziism/Fascism is one of those “hybrid” ideologies that is viewed by both the left and the right as applying to the other.
We see the left with their totalitarian top down socialist ideology as being fascist, and indeed, it is.
The left sees the conservative respect and promotion of traditional values and culture as the equivalence of the nationalism and racism of the Nazis. It’s a stretch, and they only fight it because traditional values go against their desires to embrace their sin nature:
Galatians 5:19-21
19 What the sinful nature does is clear. It enjoys sexual sins, impure acts and wild living. 20 It worships statues of gods. It also worships evil powers. It is full of hatred and fighting. It is full of jealousy and fits of anger. It is interested only in getting ahead. It stirs up trouble. It separates people into their own little groups. 21 It wants what others have. It gets drunk and takes part in wild parties. It does many things of that kind.