It would be more accurate to say that National Socialists (Nazis) were economically left and socially right. They were the "Third Way" of their era. National Socialism was a direct response to communism, fascism, and capitalism. Hitler viewed the communists as too far left and the fascists and capitalists as too far right.
The "private property" rights of the Nazi regime were largely superficial, since the government had the ability to seize private property at any time for any reason. They also controlled production to a large extent. In short, if you tried to go against whatever directives the government made regarding production or spoke out against the government, your property was forfeited to the government.
So, in practical terms, they didn't really have property rights in any meaningful sense. Collectivist policies on property are very much an economic left thing, not an economic right thing.