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I was looking for some way to describe a reversal of 'handedness' without
using mirrors. For example- two books I looked at started by saying that all
isometries could be descibed as products of reflections. Then you show that
any even number of reflections may be reduced to a rotation and an odd
number of reflections into a (reflection+rotation). Alright so far, but I
don't understand how you now classify the latter as 'orientation-reversing'.
It seemed to me that there must be more fundamental way of describing
orientation (not the specific L or R, just handedness), which you could use
to see that ther were only two forms and *then* apply that to a mirror
transformation.