Is there a way I can rotate (landscape > portrait) a digital photo without changing its (Windows XP) timestamp?

Xp has several timestamps. Are you talking about the date modified timestamps? If so then no, you can’t modify the picture without affecting that info. One option is to set your system time to the same time as the last date modified, and then perform you editing. That way the time stamp will be basically the same as before.
Let me know if this answers your question.

There are 3rd party programs for editing the timestamps. I don’t have one in mind to recommend.