It's been a very long time since I read this story, so I don't have much opinion about the text itself. However, I do have a critique of Gardner's critique, which has long bothered me, because I don't think it's quite correct.
In particle physics, the operations C (charge conjugation), P (parity, basically left-right mirror reversal), and T (time reversal) are considered separate, so there's nothing preventing us from considering a mirror-flip without turning the particles into antiparticles. Of course we have no idea what physics in higher spatial dimensions might be like. But there's no reason that a mirror-flip of a spaceship would necessarily involve C as well as just P.
How would this affect the story? Well, regardless of what Gardner says, it doesn't mean that the fourth-dimensional mirror-flip would necessarily involve matter-antimatter reversal as well. That would presumably depend on details of the fictional higher-dimensional physics that are not known to us in the real world.
But it does mean that if there is a mirror-flip without matter-antimatter reversal, it would be *detectable* to the ship's inhabitants without comparison to the outside world, through certain physical experiments involving weak-force-mediated decays.