Rotation incremental error after save

I am seeing an incremental error that is only visible after saving and reopening the file, when doing Object>Transform>Rotate.

The attached images are for the following example. Some type "hi" is written on a path (circle), starting at zero degrees on the right hand side (black, not quite visible in the pictures). Then two cases are shown:

-The black object is copied, aligned with the original, and Object>Transform>Rotate with Copy in increments (10 degrees here) up to 180 degrees.

-The black object is copied, aligned with the original, and Object>Transform>Rotate once by 180 degrees.

After that, both match at 180 degrees on screen. I save, close the file, reopen it, and there is a mismatch as observed on the attached pictures. This is for CS4.

I have tried with 5-degree increments and see less of an error, but it is there too.

Because this is rotating and comparing the same point on the circle, I suspect it is not related to Illustrator approximating circles with 4-point Bezier curves.