Oh good, because this is exactly what I was thinking when I revisited cross product definition; although handedness decides the direction of the resultant of the cross when it boils down to math, the formulas don't change, numbers are still numbers, the resulting numbers would be the same; it's humans who interpret it based on the coordinate system handedness based on our liking.
Never. In both cases the vector mathematics is the same. The only difference is that in a right handed system it will be on the right hand side of the origin, and in a left handed system it will be on the other side.
Yeah, in the reference of one system, when a new one is introduced via mirroring then this test would help.
Now, if you happen to be using either system, and introduce a scaling of -1 in one of the axes, then your coordinate-frame will now be the opposite handedness to your coordinate-system. I suspect that's what the author of the book was attempting to explain.