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I am trying to address Carl's problem in my own way but I need help.
Consider a point A located on a sphere of unit radius. The polar
axis of the sphere, z on my picture, is vertical. The two spherical
coordinates of A are TET (polar angle) and PHI (azimuthal angle).
Another polar axis, z', is chosen. Its orientation, in the old
frame, is specified by ALPHA (polar) and BETA (azimuthal). How are
new polar coordinates, TET' and PHI', expressed in terms of old
polar coordinates? I realize that the transformation is not as
simple as for xy and x'y'.