Which side are airlines curved on a world map?

Hi, you always see these pictures of a flight on a map. It's curved (that much is clear to me).

I wanted to identify the system, when the arc points south and when it points north. So I found this image:

I think it's clear to me that in the northern hemisphere, flights curve north, and in the southern hemisphere, south. If a flight were to fly directly along the equator, it would be a straight line, and if it were to fly exactly north to south or vice versa, it would also be a straight line. What I don't understand, however, is how the flights curve when a flight flies from the northern hemisphere to the southern hemisphere (or vice versa). It's always different somehow. There must be some logic, right? And why aren't the flights S-shaped then? A satellite also makes an S-shaped line on a map when it travels from north to south… (Example: ISS)