New random rant (about symmetry)

I was thinking about a universe which was symmetric about a plane (across space 3D) and without quantum randomness (ie. the future is uniquely determined by the state of each particle now).

Imagine a person near the plane, staring at himself from the other side of the universe, although with exactly the same laws. He cannot go over, because any part of him cannot possibly cross over without hitting its counterpart.

A simplistic "solution" might be to agree with himself on the other side to both stick out their right hand across the plane of symmetry. This would have worked, if in his part of the universe, right had not been left (dang!).

It would appear that this symmetry can never be broken, just like a mirror, so I wondered a few things:

1) Is it really true that this symmetry cannot be broken?
2) How do we know that the mirrors we see on the wall are not planes of symmetry in the universe (assuming the daily notions of laws of physics)?

For question 2, I realised a rather trivial solution. A key difference between a mirror and a possible plane of symmetry is basically the property about a normal mirror that it is breakable. Planes of symmetry cannot break apart just like that.

For question 1, it would appear that the relation between electricity and magnets provides a solution, because reversing both magnetic field and electric field does not reverse direction, and hence both forces can be in the same direction, but is that really true? o.O I have no idea.