I still maintain that if you cross the equator, in certain conditions, the true rhumb line track will be the same as the initial true great circle track.
We know that a great circle is a straight line on the globe, and that a rhumb line spirals towards the poles (ie spirals away from the equator). However, as the rhumb line crosses the equator, it will start to spiral towards the opposite pole; giving the rhumb line an S shape on the globe.
Try it for yourselves; I've given sample coordinates already (4500S 00000E to approximately 5800N 13554E - move it slightly around to get it accurate depending on which calculator you use.)