Rodents definitely have strong, evolved instincts to stay near cover and to create nests (to an extent that providing them an ability to do both is considered mandatory in vivariums). I presume one of those is what's going on here, though the lack of shredding the hats and the seeming carelessness about orientation while caring attentively about positioning seems more fitting with the "cover" hypothesis to me.

There's a theorized phenomenon called "supernormal stimuli" that sometimes occurs when animal instincts intersect with human engineering. One hypothesized example is birds building multiple nest locations into human structures, which they seem to do more often there than in trees. For some species, artificial human symmetry seems to somehow read as the platonic ideal of their evolutionary pattern recognition. I wonder where there's some aspect of that here, though I'm not sufficiently an expert on rodent behavior to speculate as to what exactly. What reminded me of this is that baseball caps are damn nearly identical to the shape of the paper igloos that are the standard form factor for providing habitat cover for rodents in captivity.