Your right hand looks like a left hand in a mirror, and writing on a T-shirt reads backward. Yet your head remains at the top and your feet at the bottom. Does a mirror somehow select left and right while leaving up and down alone?
Raise a hand and watch its position
Stand directly in front of a mirror and raise your right hand. Its reflection stays on the same visible side rather than jumping to the opposite horizontal location. The heights of your head and feet stay the same, too.
Why does the reflected person appear to raise a left hand? We interpret that figure as another person facing us. Their body's right and left are not the same as right and left from our viewpoint. We are mixing a position in space with the name of a side of a person's body.
The University of Illinois' explanation uses that facing-person comparison to describe the apparent opposite hand. The visible location stays consistent while our assumed body orientation changes the label.
The reflected direction is front to back
The University of Maryland's plane-mirror lecture notes describe a front-to-back reversal. For a flat mirror on a wall, the direction perpendicular to the mirror is reflected. Left-right and up-down are not independently selected and swapped.
Point a finger toward the mirror. The reflected finger appears to point out toward you. Your nose projects toward the mirror, while the reflected nose projects toward you. Those are front-back relationships on either side of the surface.
It is tempting to imagine the reflected figure as your actual body rotated around to face you. Turning a standing person through half a turn changes their left-right positions. But reflection is not the same operation as physically rotating someone. Comparing those two operations creates the familiar feeling of a left-right swap.
Why is backward writing so striking?
If you read writing on paper and then show it to a mirror, you normally turn the page so the writing faces the mirror. How you turn it affects the comparison you are making. It is easy to assign the whole change to the mirror.
A University of Maryland demonstration using a transparent sheet helps separate those steps. With transparent writing, the situation can be compared without the same need to turn an opaque page. The important question is what you are comparing and how it is oriented.
That does not mean a reflected object is identical in every respect to its original. A reflection reverses its relation to the surface; simply rotating the original does not generally make every part coincide with its mirror image. Directional objects such as letters and hands make the difference especially noticeable.
The surface is the reference
Think about the mirror's orientation rather than assuming that Earth's up and down are special. For a wall mirror, the perpendicular direction is front-back. For a mirror lying on the floor, that perpendicular direction is vertical. Reflection follows the surface, not a built-in left-right switch.
The answer: a plane mirror does not selectively swap left and right. It reflects the direction perpendicular to its surface. Comparing the result with a person facing us, or an object we have turned around, makes it feel like left-right reversal.