Single Word Requests – Is There a Hypernym for ‘Horizontal’ and ‘Vertical’?

I doubt there is such a co-hyponym (if we can call it that) to horizontal and vertical. You'll need to use an alternative name.

If you imagine the 3 axes, then the Z would appear "on the same level" as the X one. Depending on which ones you consider, 2 of the 3 will appear as such and actually, they are.

If you look at the Wikipedia page for Cartesian Coordinate System, under the section Cartesian Space it says:

For 3D diagrams, the names "abscissa" and "ordinate" are rarely used for x and y, respectively. When they are, the z-coordinate is sometimes called the applicate.

Emphasis mine. It says they are rarely used, but I doubt there are many other alternative terms, other than Z-axis, depth, and so on; they're the most appropriate terms, if you're looking for something technical.

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 The word septentrion (north) is obsolete in English, and I can find no evidence that midi (formerly spelled midy) was ever an English word at all.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.