Mod-two intersection pairing of the Klein bottle ($Q=\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)$)

A one-sided section loop and a two-sided fiber loop in the reflection mapping torus have mod-two self-intersections one and zero and meet once. Their intersection form is $Q$. In the evaluation-dual cohomology basis, the Poincare duality pairing has matrix $Q^{-1}$, giving $t^2=0$ and $x^2=tx\ne 0$.