Here’s another example of a quasimetric. My girlfriend was arguing that winter is worse than summer. Her reasoning was this: if the ideal temperature is 72 Fahrenheit, plus or minus, then winter deviates much further from ideal than does summer. In Indiana, temperatures often get down to daily highs in the 10’s, 20’s, 30’s in the winter — but they don’t get up to 110’s, 120’s, 130’s in the summer. (And that doesn’t even take nighttime / lows into account.)
But since people do choose to live in cold places, their preferences mustn’t be symmetrical. It must be that colder-than-ideal is not as bad as hotter-than-ideal. Probably because you can wear a coat, but not an anti-coat! Well, I hate wearing coats, she said.
So her preferences are more or less symmetric. But other people’s climate preferences are a quasi-metric.