Do only one or both pairs of homologous chromatids exchange genetic material during the process of crossing over?
Yes, it is perfectly possible for both pairs of non-sister chromatids to cross over in a single tetrad. The "standard textbook" depiction is a simplification, true tetrads can sometimes be very complicated and the nested crossing-over would be difficult for entry-level textbooks to cover.



This microscope image clearly shows two chiasmata occurring on two sets of chromatids, with both of the "blue" chromosome's chromatids undergoing crossing over. 

In fact, even more complicated chiasmata complexes can form during meiosis. For example, this slide depicts many chiasmata occurring in a single tetrad. More examples can easily be found by searches.