I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball. (I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that the vector field is tangential to $S^2$ at every point if you prefer the Euclidean point of view.) Is this still impossible if the hair is unoriented? By unoriented hair I mean that instead of looking at a tangent vector at every point, I am looking at a tangent line. That is, is there a rank one subbundle of $TS^2$ (which is a line bundle on $S^2$)? Geometrical intuition suggests that there is not, but that is far from proof.
I am mostly interested in $S^2$, but I expect the answer is the same for all even-dimensional spheres just like in the oriented case.