The Nazis nationalised the rail lines in the late 30s for military transport purposes, and they set up Volkswagen to provide cheap transport for the Wehrmacht and for the public as well, but other than that they did not "run many businesses." Once they got into power industries and trusts were not nationalised, indeed, military production and even film production remained in the hands of private industries. Many private companies flourished during the Nazi period while independent trade unions -- and strikes -- were outlawed. One could easly show that Roosevelt had a higher percentage of Americans on his payroll than Hitler had Germans on his.
The Nazis operated by imposing drastic fines and penalties on business owners who displeased them, usually Jews. They then invariably assigned that property to someone else who was in favor with the Reich. But the point is the property remained in private hands.
The reich was authoritarian and iron-fisted, and I've deliberately avoided using the phrase "free markets" in this discourse, instead substituting "market competition," because I accept that statist Nazi policy heavily interfered in the economy, but the type of wealth distribution policies typical of leftwing governments were never implemented by the Nazis. They wanted to put German industry onto a war footing and they needed capital to do it. That was the number one goal. Everything else took second place, including the pesky little word "socialist" in the party name.
When one considers the heavily conservative element in German politics that helped Hitler consolidate his power in 1933-34, when one understands the political calculations behind his decision to eliminate Gregor Strasser and Ernst Rohm, and when one looks at the policies the Nazis actually implemented in wielding power, it is simply not possible to call them leftwing. They were "national racialists" far more than they were "national socialists."