6PDT will work if you can get your hands on one, I've never really seen them available to be honest, but they certainly exist - might be worth noting it could well be a cheaper and less confusing (debatable) to use multiple smaller relays, ie 2 x 3PDT or even 3 x DPDT.
You'd have a circuit with a DC battery (voltage depandant on what relays you end up choosing) running through the relays and a switch inline. As for the relays, you simply wire up the two possible inputs and outputs. Here's a sample relay wiring diagram on a single DPDT for just the L and R inputs of your PC:
Note: I'm not entirely sure if this'll work with audio, If there are multiple cores carried through a 3.5mm cable, or if combining channels is the best idea, I use relays all the time for robotics but that's just to redirect power, not actual signals...
EDIT: dammit... Lizard.boy sneaked in there while I was constructing my reply