I decided that there must be female Zaxes. These female Zaxes were east-going and west-going. Every once in a long while a female east- or west-going Zax would meet up with a male north- or south-going Zax and they would do the Zaxish thing and make baby Zaxes, one or the other turning sideways as necessary.
There are lots of difficulties inherent in this. First, we still have the problem of what hapens to a male south-going Zax when he reaches the South Pole. One possibility is that he undergoes a conversion and becomes a north-going Zax on the spot (as it were). This would be a very pleasing thing for the ersatz south-going Zax, because to be a north-going Zax at the South Pole would be a fine thing indeed. It is the only time a north-going Zax has truly Free Will; he can go any direction he likes.
Other possible outcomes of a south-going Zax arriving at the South Pole might include a vast tunnel bored through the Earth from pole to pole. When a Zax reached Ultima Thule or its southern counterpart, he jumps into the tunnel and falls to the other side of the world. This was particularly attractive to me because I knew that the only place such a tunnel would be practical was pole-to-pole; anywhere else the rotation of the Earth snarls things up soemthing fierce.
If Zaxes just die when they reach the pole, we have a problem. Where do the new Zaxes come from? If a south-going Zax sires another south-going Zax, the new Zax is just as far south as his father and just as doomed. Eventually, there will be no male Zaxes left at all, and the female Zaxes will be cursed like flying Dutchzaxen, wandering around and around the globe for ever without even normal conjugal relations to ease their weary travels.
There is a solution to that, however: If the male offspring of a south-going Zax is always a north-going Zax, and vice-versa, then we could conceivably have generation after generation of Zaxes all traversing the same (relatively) narrow band around the equator. Only a few Zaxes would get too close to the poles, become trapped, and die. The biggest difficulty with this solution is that Seuss records the north-going Zax declaring unequivocally that his father was a north-going Zax before him, and his grandfather before that. This is currently the biggest mystery in the annals of Zaxology.
We're left with one other problem: what about female baby Zaxes? If they walk the same direction as their mothers, then there's trouble; one has to walk in front and the other will be constantly stepping on her heels. This is OK when the child Zax is still young, since she needs her mother's guidance and support, but when the new Zax reaches puberty, some unpleasant social problems might arise. If the child Zax goes the opposite direction from her mother, we have only postponed the problem that Seuss originally describes for the time it takes the mother-and-daughter pair to complete one-half a circumnavigation of the globe.
My guess is that female Zaxes are incubated in a pouch in the father, as are seahorses. When the new female child Zax is capable of travelling by itself, it leaps from the pouch and begins its east- or westward travel. In the meantime, the father has carried her to a different latitude.
Coming next: Social structure of the Sidehill Winders.
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