Fail. I was sooo hopeful that the video would live up to the promise of its title. Alas. I have indeed wondered for this years. I can visualize in my mind the physics, but it us hard to translate into words. And while you've done a great job of explaining what ISNT happening, you've done a poor job of what IS.
Take for instance your T shirt, "I <3 Physics". In the mirror, it's clearly reading backwards, but not because you flipped it. If you were to stand at a 90-degree angle to the mirror, with the camera at an iso angle to you both, the word would appear horizontally flipped but not vertically. It has nothing to do with your flipping it.
Of in the above scenario, you hold the arrow to your right, it will appear to the viewers to point to our left, but the mirror will show you pointing to our right.
The above phenomena are actually the z-axis flipping at work. But that explains nothing of interest. The Z axis flipping should result in the appearance of a vertical flip. And given the right angles, it does.
Imagine the mirror if place above your head, with its x-axis at 90 degrees (incidental) to the camera, but with its y-axis 45-degree downward tilt. Do we viewers see you vertically flipped, horizontally flipped, or both? The answer is both! See for instance http://upload.wikimedia.org/wikipedia/commons/f/fc/Cloud_Gate_boy_reflection.jpg
So by changing the mirror's placement and angle, we can affect whether or not the vertical axis is flipped, but we can't change whether or not the horizontal is flipped.
There is something else going on here, something not explained by "scisyhp" (nor by magic) but (I suspect) by a component of the brain.
And thus with baited breath I await from you a clearer explanation.