He may be right that Nazism has "stronger" roots on the right than I'm willing to emphasize, but that doesn't make the case that those roots are in fact strong. Meanwhile I think I do a pretty good job demonstrating that the Nazis were opposed to both traditionalism (Orthodox Christianity, Monarchy etc) as well as to Classical Liberalism. Since these two things comprise the bulk of what define modern Anglo-American conservatism, I think anatomically my argument is very strong.
However, the Nazis were also opposed to Social Democrats, who would fall into the Liberal end of the contemporary American political spectrum. And the book isn’t Liberal Nazism, which would be even more baselessly inflammatory. So we need to look at others Fascist parties, particularly Mussolini’s Fascism Classic™. The Fascist parties based their claim to undemocratic totalitarian rule on the argument that they embodied the national will. As opposed to Lenin and Stalin, who claimed the right to totalitarian undemocratic rule based on the dictatorship of the workers. Fascism was also distinguished by advocating a corporatist mixed economy, as opposed to the full bore socialism of Lenin and Stalin, with state ownership of the means of production..
Fascism and the Communism of Lenin and Stalin should be seen as the tunas and dolphins of the political ecosystem, or perhaps sharks and icthyosaurs. In any case, sharp toothed, fast moving predators competing for a similar niche in spite of some significant internal differences.
The Fascists very much defined themselves by their opposition to both Communism and Democratic Socialism, who occupied the left end of the contemporary European political spectrum. As a result their policies tended to be tailored to appeal to groups on the right, and their coalition partners tended to by from the right side of the political spectrum, during the period before they dispensed with elections. Mussolini and Franco sought church support. Mussolini maintained the monarchy, and Franco restored it. Mussolini had a count for a son in law and foreign minister. The Nazis offered industrialists protection from nationalization and trade unions and lavish government spending on munitions and infrastructure.
The Fascists were weakly socialist, but then so are and were a lot of governments most people would define as conservative. Bismark instituted a number of socialist policies, and Margaret Thatcher and Ronald Reagan left a lot of their predecessors’ welfare state in place.
I don’t thing actual fascism maps very well to contemporary American politics: anti-democracy and arbitrary rule would seem to be off the table for most Americans, let alone the piles of skulls thing. However, looking at the policies above, it’s hard to accept the notion that Fascism was mostly a movement of the left.