ballscrewbob:
It wont hurt anything to swap TX and RX over.I have an oddball Chinese one where it is as you say TX to TX it was always a pita so its in a bin of parts somewhere.
A decent picture of your adaptor may be useful in case anyone is using the same.
It's the usual PL-2303HX....
...although looking through images of these things I see though they look identical, they occasionally have different pinouts, so perhaps the ones that are TX>TX and RX>RX are actually different ones than mine.