It is possible to cross the equator twice without following a great circle, but it's not possible to follow a great circle without crossing the equator twice; p

It is possible to cross the equator twice without following a great circle, but it's not possible to follow a great circle without crossing the equator twice; perhaps this is the source of the confusion. In other words, you can reasonably say that crossing the equator is a necessary, but not sufficient, criterion for circumnavigation (it is of course not possible to cross the equator only once and return to your starting point). A good additional stipulation might be that the equatorial crossings must be separated by 180 degrees of longitude, plus or minus however tolerant you're willing to be.