Actually, this question doesn't relate to Serial Splitter. I can't be 100% sure how COM ports are being tested. However I can assume you can try connecting 2 COM ports with a null-modem cable, open them both in Advanced Serial Port Terminal and transfer data to both sides.
If there are no data loss, then obviously ports are functioning OK.
BTW, would you be so kind to describe what you mean by "to test the serial port if it works properly in TX and in RX?"?