Well. There are a few options here, and almost all of them result in:
Avoid this place like the plague
The first option is to treat the world as an infinite, flat, static plane. Your use of cartesian co-ordinates and the assumption of a 'ground' level at Z=0 implies this is true. In this case your line of symmetry must effectively be the centre of the world: Everything happens around a 'pole' of symmetry that points directly out of the ground. This is not a nice place to be, because any air that approaches this point must be moving in a radially symmetric way, effectively setting up a perfect, permanent cyclone. Whatever quirk of physics is causing this symmetry will also cause some very severe weather conditions.
The second option is similar to the first, but with plate tectonics as well. In this scenario the line of symmetry will become an area of severe, constant vulcanism as any minor plate movement must be immediately and perfectly opposed by it's counterpart, leading to immense friction and, correspondingly, heat. Couple this with the wind and you get a singularly unpleasant place to be.
The third option is to assume your world is a sphere. In this case the line of symmetry must go through the axis of rotation of the planet, or your little world
will tear itself apart in very, very short order simply cannot exist. Wind and plate tectonics are still issues. You can switch to a polar co-ordinate system and define Z as a plane passing through the equator if you want to make thinking about this easier.
I'm still working through whether or not this sort of symmetry represents a mathematical singularity or not, but if it does then you have an even more severe problem on your hands: to whit: The laws of physics break down when you reach it.
So. While pondering the awfulness that is the singularity of symmetry I realised that electromagnetism has some unfortunate things to say about this situation. I present: Amperes right hand grip rule.
This may not seem like an issue at first, however: Your world has radial symmetry. This means any magnetic field it may have must be symmetric around the pole. Now, if we assume a flat world with no meaningful magnetic field we're OK, until we get to electromagnetic interactions of molecules nearing the singularity. Things like air, or people's hands, or the ground.
In this situation any magnetic field, no matter how small, must form a curl (The magnetic field at x,y must be moving in the opposite direction to the field at -x,-y). This in turn must form a current pointing in a direction dictated by the grip rule. That must interact with the matter around it. I believe the magnetic flux becomes infinite as you approach the line of symmetry, which means the current does too. I'm genuinely not sure of my own maths/physics here (long day), so please correct me if I'm wrong.
In short, unless I've missed my maths: Nothing can reach the line of symmetry, not because it repels itself, but because the electromagnetism responsible for holding molecules together will tear them apart as they approach the mathematical monstrosity that is at the centre of your world. You have a constant line of plasma at the centre of a volcano surrounded by the fiercest winds imaginable.
Seriously. Don't go here.