Say something has cylindrical symmetry so I align it on an axis to take advantage of that symmetry (this will make things like calculating the volume of a cylinder much easier if the $z$ axis pierces through its center). I can do this because space is space and coordinate axes are arbitrary as long as we value the importance of the fact that up and down are independent to left and right and in and out spacially (and I think generally in physics?).
Let's say I choose going up to be in the $x$ direction. When I typically draw axes like this, I put $x$ along the horizontal direction, increasing to the right of the page, $y$ along the vertical increasing up, and $z$ into and out of the page increasing out. However, are these directions (up, right and out being increasing) convention? Does me making $x$ go up and down and increase up and $z$ go left and right and increasing at the right specify the direction where $y$ ought to increase by the right hand rule or something? If so, why does this violate the arbitrary-ness of coordinate axes I was talking about before?