The seeming paradox can be looked at in almost the same way as 3D perspective.

The maths is made more complicated because it is a 4D problem,

but the basic situation is similar.

Because the observers are looking at the situation from a different "angle"

they "see" things differently

A very interesting answer, but this seems to have wider implications. When mathematicians dismiss a proposition on the basis that it leads to x>y and x<y, how can they be sure the proposition does not make sense in higher dimensions. How can logic ever be used with certainty in our everyday "3D" world?

It seems to me that relativists must say that each observer, and each particle, lives in their own universe (unless they happen to have absolutely identical velocities). This way the universe belonging to each particle, and each atom in our brains, might have its own time, size, mass and clock rate. And of course each particle keeps changing its universe when its velocity changes.

This denial of a single objective world is, of course, something that most non-relativists do not share. As far as I know, experiments do not support this denial, but I'd like to learn of anything to the contrary.

The denial of a single objective universe seems bad enough, but to risk undermining the certainty of "3D" logic would be a dangerous and seemingly needless step. Science depends on logic. Can't you just say that observers have their own universes giving x>y' and x'<y ?