The demonstration is excellent except for the fact that left-right symmetry has
nothing to do with what's going on, and neither does it matter how the thing to
be reflected in the mirror is presented (right way up, upside down, etc.)
The actual transformation produced depends only on the orientation of the
mirror, though how we think about it depends on how we describe ourselves, e.g.
as having a top and a bottom, a left side and a right side, a front and a back.
One of the commentators referred to this image, which almost makes the point
we have made:
If the mirror is tilted at 45 degrees to a person's vertical axis it will
reflect a vertical person as a horizontal one, neither parallel to nor
perpendicular to the mirror.
What a mirror really does
Perhaps the most general way to think about what the mirror does is this: all
the surfaces visible from the mirror's reflective surface form a large hollow
container with a very complex shape. In producing the reflection, the mirror
turns that container inside out, something like a sock being pulled inside out,
except that a real sock may have different colours and textures on the inside
and the outside, whereas the imaginary sock turned inside out by the mirror has
exactly the same visible features on the inside as on the outside, which becomes
the inside after the reversal! This explains why the reflection of a left hand
is incongruent with the left hand, but congruent with the right hand. A
left-hand glove turned inside out fits on a right hand, and vice versa.
(This interpretation is due to Alastair Wilson.)
A familiar fact about gloves: a left hand glove will not fit on a right hand,
and vice versa, as Kant noted. Wittgenstein wrote in Tractatus 6.36111
A right-hand glove could be put on the left hand, if it
could be turned round in four-dimensional space.
But he does not seem to have noticed that even in 3-D space gloves can be
turned inside out, in which case a left hand glove will become able to fit on a
right hand, and vice versa.
It's possible that one of the commentators made these points in the many
comments on the video, but we have not searched through all of them.