I believe most of you have drawn the xyz coordinate system hundreds of times and so have I. You may have drawn it like these, on various occasions:
(the reverse directions of the axis are not shown.)
All look ok, don't they?
But if you draw like these:
then they would just look perfectly weird, because you are never supposed to see such things in real life!
Thereby arises my question: is there something to which the "visual" angles in these pictures must conform so that things wouldn't look out of place? Like an inequality or, more probably, a group of them? (well, I don't think it's likely to be equations)
Here "visual" angles take their literal meaning, say, if $\angle xOy$ looks like $120$ degrees measured from the picture, then its "visual" angle is $120$ degrees despite the fact that $\angle xOy=90$ degrees. If we denote the "visual angles" of $\angle xOy$, $\angle yOz$ and $\angle xOz$ by $\alpha$, $\beta$ and $\gamma$ respectively, then does there exist a general principle which they have to fulfil so that picture won't look visually unacceptable?
And by the way, do the "visual" angles have to do with the optics of our eyes?
I don't think there is some general principle for this. If anything, this is entirely a social construct. All of the pictures that you have drawn above are valid, and we could probably come up with various surfaces such that each is most easily visualized using each of these "visual angles".
The only thing that might take some getting used to is if you switched the x and y in these images. The reason this would be weird is because we typically choose the orientation given by "right-hand rule" for $\mathbb{R}^3$. So, if you chose the alternate orientation ("left-hand rule" I guess?) it would look strange, but it wouldn't necessarily be wrong in any meaningful sense.