I'm really trying to understand this from a mathematical viewpoint. After all, numbers are numbers, the chirality or handedness depends on how one interprets it.
Say I've three basis vectors x (1, 0, 0), y (0, 1, 0) and z (0, 0, 1). The same basis vectors would be seen differently by the two systems but the numbers remain the same.
When we do a cross product of x and y we'd get z in both systems, again the numbers of the result are the same, it's just the interpretation of the observer.
Then why do I see questions about handedness all the time? Is there a place where a game developer needs to differentiate and work in converting between the two really? Eric Haines in a related article uses the term "right-handed data". What does this really mean, aren't they all numbers? Why would they force a coordinate system on the user? Also do APIs like GL or DX force a system at some point? I've read somewhere that, in OpenGL the camera is always at origin looking at -Z direction, while it's the opposite in DirectX.
It'd be great if someone can list out where this convention really makes a difference and why.
EDIT: I think can explain it more explicitly since many seem to misunderstand the question (sorry, I'm not a native speaker of this tongue). Why do people talk about converting an asset for handedness? Say a triangle with coords (0, 0, 0), (0, 1, 0) and (1, 0, 0) would mean the same in both systems mathematically, where is the handedness here?