As a reactionary movement, they are associated with modernism in that they existed toward the end of the modernist era and were noteworthy for their rejection of it.

If you read the original essay, this question actually speaks to one of the difficulties in characterizing fascism, which is what Eco wrestles with in trying to distill it. Modernity and its love of industry and commodities were fairly core aspects of early Italian fascism, with Filippo Marinetti authoring both the Futurist Manifesto and later co-authoring the Fascist Manifesto. Contrast this with Germany, where we see more reactionary, traditionalist movements (if not straight up neo-pagan/mystic a la Julius Evola).

The root of the problem that Eco identifies is that we see something called "fascism" in both Italy and Germany, but aesthetically, ideologically, and politically these two nations look fairly different. So what exactly fascism is becomes difficult to pin down and taxonomize. If I'm not mistaken, I think fascism isn't not strictly modernist, and mostly rejects modernity in favor of traditionalist structures. The aesthetic modernity associated with German and Italian industrialism derives either through their love of technology as a means to realizing an imperialist and nationalist ideology, as well as a sort of hyper-masculinity associated with weaponry and militarism that feed into fascism's constant hero-making.

Well, it's a relatively new label for ancient techniques.

Same shit, new package, so to speak.

I think fascism is often thought of as reactionary backlash against modernity.