From MTW (there's a PDF version on line, wjhich I acquired after originally purchasing a hardcopy of fthe book):
Why do half-angles put in an appearance? And what is behind the law of combination
of rotations? The answer to both questions is the same: a rotation through
the angle 0 about a given axis may be visualized as the consequence of successive
reflections in two planes that meet along that axis at the angle 0/2 (Figure 41.2).
Two rotations therefore. imply four reflections. However, it can be arranged that
reflections no. 2 and no. 3 take place in the same plane, the plane that includes
the two axes of rotation. Then reflection no. 3 exactly undoes reflection no. 2. By
now there remain only reflections no. 1 and no. 4, which together constitute one
rotation: the net rotation that was desired (Figures 41.3 and 41.4).
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x
Figure 41.3.
Composition of two rotations seen in terms of reflections. The first rotation (for instance, 90· about
02 in the example of Figure 41.1.) is represented in terms of reflection I followed by reflection 2 (the
planes of the two reflections being separated by 90·/2 = 45· in the example). The second reflection
appears as the resultant of reflections 3 and 4. But the reflections 2 and 3 take place in the common
plane 20X. Therefore one reflection undoes the other. Thus the sequence of four operations 1234
collapses to the two reflections I and 4. Their place in tum is taken by a single rotation about the axis
OA.
Neat!
I'll repost my original, which explains how this is relevant to my query, but with some modifications,
Thanks for your interest..