I’ve learnt that in crystallography, there are many crystallographic directions that are “equivalent.” For example, in a cubic crystal system, the  and  and  directions are “crystallographically equivalent.” See the image below:
I cannot begin to fathom how they are equivalent. These are vectors after all, and from what I can tell with my rudimentary knowledge of maths, these vectors are not equivalent in direction. Yet all materials science textbooks and powerpoints insist that, for cubic systems, any permutation of positive and negative indices will yield a same equivalent direction. That is, the , , and  vectors are… the same vector?
I share the same confusion with planes, even if they aren’t vectors. For cubic systems, just as for directions, any arrangement of positive and negative indices produces an equivalent plane. For example, (100) and (010) are the same plane, when to me they clearly look different.
So what exactly is “crystallographic equivalence”?