It occurred to me that latitude and longitude are not homologous in that the lines of latitude lie on parallel planes or slices through Earth, while lines of longitude are not parallel and meet at the poles, forming segments through Earth. Obviously the Earth's rotation gives us two points that make sense to use as intersection points, but it struck me that there could be potential advantages for using arbitrary points for intersection points for longitude, or for disregarding the poles for latitude. Not knowing the names of these I was not able to find any information online.
Is there a name for this difference, or names for the two types? Are there any coordinate systems that use parallel longitudes (or non-parallel latitudes) and what would the implications of this be?