Regional or directional references that are almost universal in US usage include those two and others. After a childhood up north, I moved out west, but my relatives are mostly back east, except for a sister who is currently working down south. Deviations from these phrases, like out east, are rare, bordering on strange.
I assume that up north and down south relate to a map held with north at the top. The upper peninsula and lower peninsula of Michigan are so named because of latitude, not altitude. Going out west comes from the frontier drive toward "out there", I suppose, and once out there people recalled the time when they lived back where they came from, a bit like the 19th century immigrants who referred things done back in the old country (where they came from, Germany, Italy, etc.). But even an immigrant to the west coast from Mexico or China, for example, might refer to New England as being back east, once on top of AE. Over is for over seas, literally, like over there in France.
In BE, if I have it right, students go down to London from Cambridge, but a different twist might be involved here, as someone might also go up to London.
In German it seems that upper and lower parts of Bavaria (and "Saxony"?) relate to altitude or maybe to being down stream. Are there other such fixed phrases for regions or directions that are at least regionally common and that an outsider might not be familiar with?