What Mirrors Do

Why do mirrors reflect left-right, but not up-down?  Suppose we have 
a card with the letters

                          b  q
                          d  p

printed on it (as viewed from the front of the card).  If we are
going to view this card in a mirror we need to turn it so the front
of the card faces the mirror.  What we see in the mirror will then
be the same as what we would see if you could view the printed 
letters from the back of the card (e.g., by holding it up to the 
light.)

But what we see will clearly depend on HOW we turned the card "over".
If we rotated it about the vertical axis we will see

                         p   d
                         q   b 

i.e., it will be reversed left-right, but if we rotate it about the
horizontal axis we will see

                         q   b
                         p   d

i.e., it will be reversed up-down.  This applies both to what we see 
in the mirror and what we see through the back of the card.  In fact, 
we can dispense with the mirror altogether and just use a transparent 
plate with an image on it.  We're free to rotate the plate about any 
axis (parallel to the plate) we wish in order to view the image from 
the other side.  The axis of image-reversal will correspond to the 
axis of rotation we choose.

Of course when looking through a mirror at, say, our living room, we 
don't physically rotate the room.  Instead, we re-orient ourselves,
looking first at the room (the reference image), and then TURNING to 
look at the room's image in the mirror.  To execute the turn we choose 
an axis of rotation which is usually vertical, resulting in a left-
right image reversal.  On the other hand, if we look at the room and 
then rotate about our horizontal axis to look at the mirror, the image 
we see will be reversed up-down relative to what we saw directly 
(because we are now standing on our heads).

There's actually an extensive literature on the question of "what
mirrors do", and it has been discussed from many different points of 
view.  It's generally accepted that the question is ambiguous because 
the entities (visual images) and operations (reversals) to which it 
refers are both inherently interpretative and of a higher conceptual 
order than the simple concept of a mirror.  We can say without 
ambiguity that a mirror reflects incident rays of light, and the angle 
of reflection equals the angle of incidence, because the concepts of 
rays and angles are of the same order as mirrors.  However, the 
notions of "visual image" and "reversal" are of a higher order, and 
their precise meanings are a matter of interpretation.

For example, a "visual image" may be regarded as a two-dimensional 
entity (which, in a purely geometric-optical sense, it is).  In this 
sense the "effect" of a mirror can be defined as the mapping it gives 
between a 2D visual image viewed directly and the "same" 2D image 
viewed through a mirror.  In these terms, the "reversal" seemingly 
effected by a mirror is really just a consequence of the rotation 
performed by the viewer as he re-orients his field of vision from 
the direct to the reflected image.  These two fields are necessarily 
oriented at 180 degrees from each other, but this doesn't fully 
contrain how the field of vision is adjusted, because the field of 
vision consists not only of a direction in space but also an 
orientation about that direction.  

Thus, if I first view an image due North, and then view the same 
image through a mirror due South, the axis about which the image will 
be reversed is precisely the axis about which I rotated my field of 
vision as I turned from North to South.  If I turned about a vertical 
axis (as is most likely) the image will be reversed left-to-right, 
but if I turned about a horizontal axis (by standing on my head, for 
example) the image will be reversed top-to-bottom.

On the other hand, a "visual image" can also be construed as the 
3D model that we psychologically associate with a particular 2D 
optical image.  In support of this notion we can point out that
our visual images actually consist of input from TWO eyes located
at slightly different positions, so there is some justification 
for including "depth perception" as an inherent aspect of a "visual
image".  On this basis people sometimes say that the effect of a
mirror is to reverse front-to-back, i.e., things that were furthest 
South in our 3D model of the direct view will be furthest North in 
our 3D model based on the reflected view.

However, although this approach seems plausible (and in fact it's the 
answer given in the sci.physics FAQ), there are several problems with 
it.  On a purely formal level it can be argued that a visual image 
(singular) is properly defined relative to just a single point of 
view (eye), and that the human visual visual sense (for people with 
two eyes) is composed of two distinct visual images, from which we 
psychologically synthesize a mental image.  Thus, we would argue
that the 3D approach is answering a different question than the 
one that was asked (i.e., it's dealing with mental images rather 
than optical images).

More seriously, identifying the notion of "visual image" with the 
corresponding 3D model has a serious problem of non-uniqueness.  It's 
well known that many optical images are ambiguous as to their 3D 
interpretation.  For example, consider the image

                         /|\
                        / | \
                       /  |  \
                      /   |   \
                     /    |    \
                    /  *  |     \
                   /     / \     \
                  /    /     \    \
                 /   /         \   \
                /  /             \  \
               / /                 \ \
              //_____________________\\

Is this image convex or concave?  In other words, are you looking
at the top of a pyramid, or down into a triangular-shaped hole?  
Obviously this is just a 2D ASCII image, but there are real 3D 
objects and lighting conditions that present images just like this.  
If you look at such an object in a mirror, will it be reversed front 
to back?  What if you interprete it as a pyramid when viewed directly, 
but as a hole when viewed through a mirror?  In that case you would 
have to say the mirror did NOT reverse the image front to back.  
Nevertheless, the asterisk will appear to be on a different face, 
so clearly the mirror has done _something_.  How can we characterize 
the effect of a mirror in a way that doesn't depend on subjective 3D 
interpretations?

All we can say for sure is that, in the 2D sense of pure optical 
images, an image is reflected about the axis around which our field 
of view is rotated as we turn from the direct view to the mirror.  
This is adequate to explain all the "reversals" that occur, including 
reversals of "handedness".  For example, the image

                      * * *
                          *

will appear as some plane-rotated version of

                          *
                      * * *

when viewed through a mirror, but no plane-rotation will make it
look exactly like the original, because one is left-handed and the
other is right-handed.  Thus, the notion of "handedness" doesn't
require three dimensions.  The change in handedness produced by
mirrors is fully represented in the 2D optical approach.  Furthermore, 
the 2D approach is absolute and unambiguous.  In contrast, any 3D 
"front-to-back" effects that we may attribute to a mirror are 
necessarily based on ambiguous psychological interpretations.

For related discussions, see
A Mirror To Physics
Reflections on the Mirror Question