connecting one serial cable to the other, and running two instances of HyperTerminal on the two different ports will verify whether or not it works correctly.
but the thing to remember is both of your com ports are acting as a DTE ... in order to communicate straight through, one has to be a DTE and one a DCE. the difference between DTE (data terminal emulator) and DCE (data communication equipment) is that the TX and RX pins are swapped (pins 2 and 3, on a DB-9)
therefore you need a "null modem" cable to communicate between two com ports on two DTE machines (ie, typical PCs). In other words, cross over your pins 2 and 3 and retry :-)
here's a good reference
since you're not using handshaking, your task is much easier. if you can get both of the COM ports to communicate back and forth -- in each direction -- then the problem is not your USB-Serial converter. If it's not the hardware, and you're certain your code is correct, then I would think its a Windows application setting. Perhaps your API limits the default Com Ports to COM1 - COM4 ?
if you get a chance,
try using TeraTerm. it's open source freeware, and highly configurable, and IMO much better application than HyperTerminal. I think you'll like it.