Were you educated entirely in the US?
I think that's the biggest issue when it comes to landing on definitions - the modern US political spectrum is deeply skewed, and often totally at odds with the longstanding definitions in use pretty much everywhere else. There is some sort of pathological need to try to up-end traditional definitions and force Fascism and Nazism to the left wing as a political 'gotcha', typically by cherry picking various bits and pieces.
Generally, outside of historical nuances of the left being common landowners and the working class, while the right was represented by landed gentry and traditional aristocracy, the political science approaches have mirrored the classic left and right economic axis, with common-ownership, worker centric economics on the far left (Communism), and private but state controlled, corporatist economics on the far right (Fascism). This is a reflection of it originally being a scale of radicalism versus traditionalism, with the 'modern' (as in, no monarchy or aristocracy) extremes being a stateless, workers revolution on the far left, and a rigid, totalitarian corporatist traditional state on the right. Fascist leaders explicitly self identified as being on 'the right'.
It's the two axis model which then more directly introduces measures of authoritartianism or anarchism into the equation.
I think the Nolan Chart is probably the most useful measure these days.
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Huge amount to unpack here, but no. This statement is about as inaccurate as it gets, which should be pretty self-evident to most with even a basic understanding of the competing and opposing ideologies involved.
Praising NSDAP policies for addressing issues plaguing German workers in the wake of the First World War and Great Depression is hardly evidence to suggest the two ideologies are working in lockstep, especially and particularly when Hitler is busy writing about his plans to exterminate "Marxism" from Germany, and working in tandem with the traditional German right against the left. Even prior to actively persecuting via violence, the abolition of trade unions and forced exile of left-wing civil servants and academics paints a pretty clear picture.