<--(American) right wing-------------- (American) left wing--(european) socialism--fascism-communism-->
Anyone who thinks fascism is right wing (american right), is not worth wasting any of your life talking to. There is a tiny handful of issues that might fall between the fascist realm and american right. Usually minor social issues with little bearing on the overall government. There is a gigantic mountain of issues the left and fascists are lock step in.
The modern American political spectrum does not define left vs right. Left vs right was a term which originated in the 19th century to describe the difference between conservative monarchists who were right wing and liberals who were left wing. That is reductionist, but really the entire idea of left vs right is reductionist in and of itself so whatever.
In that set up, the right was the side of big government, support for the status quo, and support for hierarchy. The left was the side for reducing government, opposing the status quo, and opposing ingrained hierarchical structures. This was the way it was conceived for a long time, and this continued to be the way it was conceived in much of the world until WWI, when most of the monarchies of Europe fell. It was in the aftermath of this period that fascism arose. It was a set of beliefs which supported big government, hierarchy, and many other things that were staples of conservative and right wing movements (like nativism, militarism, etc). Their ideological aims were antithetical to everything Marxists believed in. They cared little about class based struggle, equality, etc. They cared about the promotion of the nation and the national citizen. At their core that it what they were, a nationalist movement. An extreme form of nationalism. They often supported a contradictory set of polices to promote this goal. They supported capitalism, but with heavy state intervention (something capitalist/liberal countries do in times of war as well, but Nazis thought treating an economy like constant war times was a good idea). In the end they didn't have strong ideological convictions when it came to economic polices, and instead supported a piecemeal economic policy that placed practicality and expediency in the ascendance. They created a coalition government with a conservative/right wing party when they won their one and only election. The first concentration camps were set up to imprison leftists, not anyone else.
However having said all that, Nazis don't fit in neatly on a modern American left v right spectrum. That is because old left v right spectrums had liberalism on one side and other ideologies in opposition. First monarchism, which placed liberalism on the left. However after the end of monarchism as a major force in the "West' you saw the rise of a new spectrum, which put liberalism to the right of communism. Then the collapse of the Soviet Union occurred, and you no longer had ideological opposition to liberalism. Instead you had debates within the liberal spectrum. So now the left vs right spectrum is essentially between classical liberals on the right and social liberals on the left (again reductionist, but as I said, you can't do this left v right **** without being reductionist). Since Nazism was an entirely illiberal ideology, it DOES NOT fit on an American left v right scale. Nazism had qualities and goals that it shared with the modern American left AND the modern American right. It is why you can have maniacs like Glen Beck tell you the left is Nazi, and then point to similarities, and the left do the same and point to similarities. When in reality neither side are Nazis.
PS. Statism is an idiotic barometer and measure for a left v right spectrum. All major modern political movements, without exception, have seen the state as the primary social mover. Belief in the positive power of the state is something shared by liberals (both the classical and social kinds), communist, and fascists. Only Anarchists and libertarians disagree, and they have never had enough popular support to actually do anything of consequence.