Mod-two cohomology ring of an inversion mapping torus ($H^{\ast }(M_n;{\mathbb{F}}_2)={\mathbb{F}}_2[t,x_1,\dots ,x_{n-1}]/(t^2,x_i^2+t{x}_i)$)

The base class $t$ and coordinate classes $x_i$ all have degree one. Pullback of the mod-two intersection pairing of the Klein bottle gives $x_i^2=t{x}_i$. The Leray-Hirsch theorem gives the basis consisting of square-free products $x_I$ and $t{x}_I$. Thus the additive dimensions match those of a torus, but the degree-one squares differ for $n\ge 2$.