No Tight Klein Bottle
If $\alpha(f)$ is the number of top cycles for a tight immersion of $M$ (not a sphere), then
2\le\alpha(f)\le2-\chi(M).
For the Klein bottle:
$\chi=0$ so $\alpha(f)=2$,
$M^+$ is a cylinder,
$M^-$ is a cylinder that reverses the orientation of its boundary, a contradiction.