In Reid's compass experiment, why does the compass rotate one way, then another?

NOTE:  This is a first guess at an answer.  Maybe Reid can chime in with a REAL answer (!)

As shown in this illustration (credit Peter.reid@ed.ac.uk), the Earth's magnetic field loops into space.  Reid's compass, which is mounted vertically, shows the curvature of these lines.  When ISS is in the northern part of its path, the compass follows the lines of the curve, pointing mostly down.  If it were directly over the north pole, it would point straight down.

The opposite effect takes place in the southern part of the ISS path.

And as it crosses the equator, the magnetic lines (and compass) are horizontal.

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