In my world, there are two types of people, lefties and righties. They have opposite orientations, which means they are mirror images of each (they are Enantiomers of each other essentially). Throughout much of history, most people have been "orientist", or "handist" (since lefties are left-handed and righties are right-handed), meaning that they hate those of the opposite orientation. In each country, usually either the lefties or the righties were in power, oppressing harshly those of the opposite orientation (usually enslaving, but sometimes killing them, and sometimes letting them be mostly free but limiting their rights).
In my country, for example, the righties are in power. Most of our neighbors are also pro-rightie, but one of our neighbors was taken over by a foreign leftie civilization 150 years ago. We have only recently established a tenuous peace with them.
The idea of orientation has an interesting history. About 300 years ago, a foreign leftie astronomer actually came up with the idea that there is no absolute notion of orientation in our world (i.e. it was non-orientable). He was promptly put to death by his government for such an absurd notion.
About 200 years ago, a number of linguists from different countries found a rather odd linguistic paradox. If you translate "rightie" from English, through seven different languages, and then back to English, it comes out "leftie". If you start with "leftie", you get "rightie". The linguists were quick to point out that language is a very fuzzy thing, but some suspect them of non-orientist sensibilities.
Nowadays, very odd things have started happening. The world now has a global economy, so people and goods travel around the world. World travel has the disastrous effect of sometimes corrupting the orientation of goods, or even people, to the opposite orientation! Some scientists are even asserting that the astronomer was actually right, and our world is topologically a Projective Plane!
How should we respond? Everyone knows that lefties are inferior. How can we justify oppressing them if orientation is non-stable? How should we treat world travelers, when we don't know their original orientation? (That is, how can orientists react and adapt to globalization in this world? Is there a way they can keep their prejudice?)
- Non-orientability means that you can only compare the orientation of objects that are near each other. There is no consistent way to define orientation globally. It also means that if you travel around the world in a certain way, you become mirror reversed (compared to, say, a sculpture someone made of you before you took the journey which didn't move). See this picture to see what that looks like.
- Orientation (locally) is very easy to tell. Besides handiness, there are also very distinctive biological differences in appearance that are very hard to cover up. (Attempting to cover them up is considered taboo.) There are few other physical differences though. (One of them is that are a couple of foods that only righties or only lefties can eat. This has led to unfortunate effects in the more genocidal nations.)
- There are many equivalent ways to define the projective plane, but one way is the hemisphere model. In this, the projective plane is a hemisphere, in which opposite points on the rim are glued together. This means that if you walk of the rim, you pop up on the other side, but with your orientation reversed. (You might be tempted to say that this is the answer to my question. Just define someone's orientation as what their orientation is at birth, in the hemisphere model! There are two problems with this: (1) For a given projective plane, there are infinitely many different hemisphere models. The orientists would have to pick one, and justify why that one is the best. (2) Perhaps even bigger, orientists near the "rim" of the chosen hemisphere model would regularly change orientation (according to the model). They probably wouldn't accept such a model.)
- Orientation is inherited from the parents. This means that if your parents have the same orientation at the time of conception, you will have the same orientation as your mother at birth. If you parents have different orientations at the time of conception, you will have a random orientation when you are born. (Also, you will probably be killed.)
- So, I've only described the topology of the world. Allow me to now describe the geometry: Establish a 3D coordinate system, where the planet's center is at (0,0,0). Points (x,y,z) and (-x,-y,-z) are the same point, and otherwise physics is mostly normal (except for how the sun works). This means that the planet's surface will topologically be a projective plane. It is also about as round as the earth. There is a star which goes around the planet. It cycles between light and dark every 12 hours (don't ask me how). (The reason this is necessary is the equator only has 180 degrees. This means the sun is always visible. If the sun was always light, there would be no night.)
EDIT: In the comments, many people are suggesting that a solution would to simply "ignore" the issue. (Note: Before globalization, ignoring the problem was a valid solution, and what the people did.) Although for some social issues that works, it doesn't in this case. Here are some scenarios where the ignoring it won't work:
- A prominent foreign person comes to town for some reason (business, politics, acting, etc...). They appear to be a rightie, but there are rumors that they where born a leftie. How do you decide whether or not to hate them?
- The rightie association of this country is having a video chat with the rightie association of some far away foreign country. To their great surprise, when the chat starts, they appear to have opposite orientation (i.e. they appear to be lefties to each other). One of the IT people fiddles in the settings, saying "okay, I swapped the video orientation". They now appear to be the same orientation, but both associations stare at each other suspiciously for a couple minutes, wondering if they are actually the same orientation or not.
- A rightie sends his rightie kid away to college. When they come back for Christmas, they are leftie. They did not decide to be leftie; it was just a side effect of the route their plane chose.
As you can see, this is very different from the flat v.s. round earth issue. This can actually affect people personally (if they care about orientation, that is).