What emerged as the fascist economic system then was a combination of the socialist rejection of capitalism and the nationalist rejection of internationalist socialism. It’s not coincidental that “Nazi” was short for National Socialist German Workers Party. The very name suggests that the fascists started from a socialist premise (including the emphasis on being a “workers” party), but added the “nationalist” (and specifically “German”) twist.Avraham Barkai writes in his book "Nazi Economics: Ideology, Theory, and Policy", Oxford: Berg Publishers Ltd., 1990.
Rather than have full-blown socialism as we saw in the early years of the Soviet Union, the fascists generally preferred hybrid forms that often maintained the appearance of elements of capitalism but with a much larger role for the state in allocating resources. A look at the Nazi Party platform of 1920 shows the very strong influence of socialism in the economic planks, including objections to the earning of interest, the desire to nationalize industries, the confiscation of profits, and land reform. Not all of these were put into place when Hitler gained power, but the Nazis’ antipathy toward capitalism is quite clear, even as they often co-opted big business into their power structure in during their reign. The trappings of private ownership were often preserved, but the Nazis used the power of the state to try to ensure that private ownership was used as a means toward the national ends that they defined.
The Italian model was similar in its broad outlines, though different in its execution. The Italians were more clear than the Germans about the way in which market competition was destructive of national goals. They didn’t see Russian socialism as a solution for the reasons noted above. Instead, they argued for industry-level partnerships among labor, capital, and the political class. The idea was that by working collectively, these cartel-like organizations could resolve questions of what to produce, what price to charge, what wage to pay, and the like all without the need for cut-throat competition among firms or workers, or the use of strike threats between workers and capitalists. By putting national interests first, these collectives could plan out production industry by industry and ensure a cooperative peace among Italians. So, once again, the system kept some of the trappings of capitalism, such as nominally private ownership, but set them in a system where collective planning of a limited, and nationalistic, sort was the overarching structure.
Both of these systems are probably most accurately called “corporatism.” In such a system, we get these sorts of private-public collaborations in which private ownership is combined with state control and privileges for labor, and where all are expected to serve some larger national goal. It looks like private ownership, which is often the source of the claim that fascism is a form of capitalism, but the degree of distrust of the unplanned order of free markets and the de facto power that falls into the hands of the state to set goals both point to it as being more accurately a form of socialism or planning.
In an off-the-record talk with a newspaper editor in 1931, Hitler defined the basic principle of his economic project: "What matters is to emphasize the fundamental idea in my party's economic program clearly-the idea of authority. I want the authority; I want everyone to keep the property he has acquired for himself according to the principle: benefit to the community precedes benefit to the individual ["Gemeinnutz geht vor Eigennutz"]. But the state should retain supervision and each property owner should consider himself appointed by the state. It is his duty not to use his property against the interests of others among his people. This is the crucial matter. The Third Reich will always retain its right to control the owners of property." Here we have all the basic assumptions that later determined the Nazis' economic plans and their actual policy: virulent antiliberalism, subjection of the economy to the primacy of political and social goals as defined by the national leadership, and state control over all economic activities. [p.26-7]So you could do anything you liked with your own property, just as along as it was exactly what the state wanted you to do with it.