All points in a perfectly flat mirror will be reflected directly back in relation to their original position.

The 'left-right' reversal is only in the mind. The mirror doesn't 'do' anything except show a.....well, a mirror image. The 'left eye' think you see in the mirror is actually the right eye. Imagine a photographic negative that can be viewed from either side. Make a dot on the left eye, then look at it from the back. The dot hasn't moved just because you now see it 'on the other side'...and turning the negative upside down doesn't change the relative positions of anything.

All this is why, in telescopes and binoculars and cameras, extra lenses are used to reverse images a 2nd time so that we DO see left & right the way we would 'normally'.