You can of course comb the hair around a circle, in one of two directions, clockwise or counterclockwise. You only get into trouble on the sphere. A more general proof addresses spheres of all dimensions, but this proof is specific to S2 .
Put the sphere in front of you with the north pole at the top and the south pole at the bottom. Let arrows represent the tangent vectors, i.e. the direction of the hair. Walk around the equator heading east, always looking inward toward the center of the sphere, and note the direction of the arrows as you go. The direction moves continuously. It may jiggle and jitter about, and even spin around, but when you get back to start on the equator, the arrow has returned to its original direction. This is a continuous map from the circle (equator) into the circle (arrow direction).
Let the winding number be the net number of times the arrow spins around clockwise as you travel around the equator. This has to be an integer. If the arrows point east, for example, all the way around, the winding number is 0. If the arrows point north all the way around the winding number is 0. More likely, the arrows will move this way and that as you go, and perhaps create a net spin when you get back to start. The winding number is an element of the fundamental group of the circle.
This can be done for any latitude, other than 90 or -90 degrees, which are degenerate cases. If l is the latitude, walk around the sphere at latitude l and note the winding number of the arrows at that latitude. Call this w(l).
The arrows are continuous across the surface of the sphere, and that means they change hardly at all between 51.23 degrees and 51.24 degrees, for example. More formally, w(l) is continuous in l. Since w(l) is an integer, it can't continuously jump from one number to another. Therefore w(l) is constant across all latitudes. Seen another way, the arrows from one latitude to another implement a homotopy, and that cannot change the winding number, which is an element of the fundamental group of the circle.
Continuous on a compact set is uniformly continuous, so there is some little circle about the north pole, say 89 degrees north, where the arrows don't vary by more than 1 degree. They are all pretty much in line. If the arrow at the north pole points right then everything above 89 degrees points right. This determines w(l). March around the circle at 89 degrees north. First the arrows point right, which is east, which is the direction of travel. But a quarter of the way around the circle the arrows point south, relative to the circle being traced. On the back side the arrows point west, then they point north, then east again. The arrows spin around once clockwise, hence w(89) = 1, hence w(l) = 1.
Do the very same thing at the south pole, and w(l) = -1. The arrow spins around once counterclockwise as you traverse the equator. Since w(l) cannot be both 1 and -1 simultaneously, the continuous field of arrows cannot exist.
If you delete the south pole then the problem goes away. w(l) can equal 1 for -90 < l < 90.