How does one correctly form the "north" and "south" forms for which occident and orient are "west" and "east"?
I found boreal and austral, but those look like adjectives and I'm after the nouns.
Bonus: what about "up" and "down"?
The words orient and occident are two of the set of six French words
orient, occident, zénith, nadir, septentrion, midi,
which form the set you were looking for. The word septentrion (north) is obsolete in English, and I can find no evidence that midi (formerly spelled midy) was ever an English word at all.
In Old French, the word méridien was used instead of midi (see wikipedia), so another possible sixth term is meridian, which is indeed an English word which has occasionally been used to mean the opposite of septentrion; see for example this reference from Google books. So maybe the best answer is:
orient, occident, zenith, nadir, septentrion, meridian.
You shouldn't expect anybody to understand the meaning of the last two terms without explanation, though.
I think boreal and austral may be the closest you'll get.
According to Etymonline, The noun form of austral is an English word:
auster "south wind," late 14c., from L. auster "the south wind; the south country" (see Australia).
However, the noun form of boreal, is only the Latin for north wind, boreas, or the name of the Greek god of the north wind, Boreas.
In Portuguese the words are "meridional" and "setentrional", both with roots on French. I checked on-line and saw that "Meridional" and "Septentrional" are valid English words.
North and south can both be nouns, enabling us to speak of the North and the South, just as we speak of the Orient and, perhaps less often, the Occident. To be consistent, we'd then have to speak of the East and the West.
Perhaps the closest you might get to a modern correspondent to "austral" is "the Antipodes" (and adjective Antipodean).