The fundamental group of the Klein bottle is $D_\infty$, the infinite dihedral group (which is $\mathbb Z \rtimes \mathbb Z_2$).
I believed this for some time, and I seem to recall some others having the same confusion.
The group that has been mistaken for $D_\infty$ is in fact $\mathbb Z_2 *\mathbb Z$, though it is usually written with the presentation $x^2y^2=1$ or $aba^{-1}b=1$. They both abelianize to $\mathbb Z\oplus \mathbb Z_2$, which i think was part of my confusion.