So, before attempting to make a cable myself (I'm pretty uneducated in that sort of thing), I tried the Belkin Model F3U200-08INCH 8" Joystick Adapter DB15F/4Pin USB A M F-M and a gender changer to connect the Quasicon to my PC via USB. And it didn't work! I'm trying to work out how/why that combination of wouldn't work, and I'm open to suggestions. Is it possible Quasimoto's cables were more complicated than originally suspected?
Similarly, I tried the PS2 connection that came from Quasimoto and a PS2 USB PC adapter, which also didn't work, but came a little closer - the lights on the Quasicon at least came on, indicating that it was getting power, and the PC recognized it as a USB controller, but it didn't recognize any button presses or stick movement. And yes, the Quasicon works perfectly when plugged into a PS2.
I'm confused by these results, clearly. If this isn't the appropriate forum for this sort of question, I apologize and will stop pestering you all.