Why are mirror images flipped horizontally but not vertically?

Here's a video of physicist Richard Feynman discussing this question.

Imagine a blue dot and a red dot. They are in front of you, and the blue dot is on the right. Behind them is a mirror, and you can see their image in the mirror. The image of the blue dot is still on the right in the mirror.

What's different is that in the mirror, there's also a reflection of you. From that reflection's point of view, the blue dot is on the left.

What the mirror really does is flip the order of things in the direction perpendicular to its surface. Going on a line from behind you to in front of you, the order in real space is

1. Your back
2. Your front
3. Dots
4. Mirror

The order in the image space is

1. Mirror
2. Dots
3. Your front
4. Your back

Although left and right are not reversed, the blue dot, which in reality is lined up with your right eye, is lined up with your left eye in the image.

The key is that you are roughly left/right symmetric. The eye the blue dot is lined up with is still your right eye, even in the image. Imagine instead that Two-Face was looking in the mirror. (This is a fictional character whose left and right side of his face look different. His image on Wikipedia looks like this:)

If two-face looked in the mirror, he would instantly see that it was not himself looking back! If he had an identical twin and looked right at the identical twin, the "normal" sides of their face would be opposite each other. Two-face's good side is the right. When he looked at his twin, the twin's good side would be to the original two-face's left.

Instead, the mirror Two-face's good side is also to the right. Here is an illustration:

So two-face would not be confused by the dots. If the blue dot is lined up with Two-Face's good side, it is still lined up with his good side in the mirror. Here it is with the dots:

Two-face would recognize that left and right haven't been flipped so much as forward and backward, creating a different version of himself that cannot be rotated around to fit on top the original.

Because they don't flip left with right (or up with down), they flip the 3D space you're standing in "inside out", so far from the mirror becomes far away inside the mirror and vice versa. A hand 1 meter from the mirror seems like it's 1 meter on the other side of the mirror but in the same spot with regards to left/right so nothing is flipped.

Wiggle your left hand - you'll see the hand which is to the left in the mirror wiggle. Wiggle your toes and the toes in the mirror image wiggle etc.

This common confusion stems from our familiarity with photographs. We forget that we rotate them to face ourselves.

Take a picture of yourself and hold it up in front of you. Probably you are holding it so that you can see your image. If so, you "flipped" the image of yourself when you rotated it 180 degrees around the vertical axis. When you look to the left side of the photo, you are looking over the right shoulder of your image. These directions are flipped!

Now look in a mirror. When you look to the left, you are looking over the left shoulder of your image. These directions are not flipped!

Now pick up the picture again and turn it so it's facing the same direction you are facing. You have removed the 180 degree rotation so that you and your image are "looking" in the same direction. The left side of your image is again to your left. If the picture is transparent enough that you can see your image, you'll see not the back of your head, but your eyes, giving you the impression that you're looking back at yourself. A mirror image! But again, left and right are not flipped.

When you say the mirror "flips" left and right, you are speaking from the frame of reference of one who is used to the 180 degree rotation that you apply to view an opaque photograph. But that's what we all do because we consider photographs, rotated 180 degrees to face ourselves, as being the "correct" left-right orientation.

What a mirror really flips is the depth dimension. That which is behind you appears to be in front of you.