Katie,

In general, the product of two rotations (with different centers) is another rotation. Sometimes it turns out to be a translation.

So, for example, two half turns about distinct centers gives a translation (by double the vector between the two centers).

The product of two rotations by 90 degrees turns (clockwise) out to be a half-turn about a third center (form a 45, 45 90 triangle using the first two centers to find the third center).

The best way to investigate the products of rotations is to represent each of the rotations as the product of two reflections (in mirrors through the center of rotation, with an angle 1/2 the angle of rotation). Then there is a system of combining the mirrors to find the mirrors representing the product. If the final two mirrors happen to be parallel - it is a translation. Otherwise they intersect at the new center of rotation (and make an angle 1/2 the angle of rotation of the product).

The same principles apply on the sphere, but of course the mirrors always intersect and there is no translation!

Walter Whiteley
York University