Yes, a cable can be re-used after it's been unplugged from one set of devices. It's basically copper wire (actually 4 pairs)... When you un-plug the RJ-45 connectors (those fat connectors you see on the ethernet cables), the cables are still good. Cables can go bad (if for instance they got a really bad kink in them and the wires inside got crushed, etc), but simply moving from one device to another doesn't pose a problem.
Basically, you want to have 1 cross in the path between your 2 devices somewhere, else an odd number of crosses. What it's doing is crossing the transmission line on one device, to the receive line on the other (aka the orange and green pair). It needs to do this, so each respective device can receive the "messages" sent to it by the other...
Generally, when your connecting 2 similar devices you need a cross. This would be PC to PC, router to router, switch to switch, etc. PC to router is the same sort of deal.
In the case of hubs and switches, they generally introduce a cross inside the device. This is why you use a straight through, when connecting the PC or router to a hub or switch. If you used a cross over cable connecting a PC to a switch for instance, it would be like connecting 2 cross over cables in series, hence no cross in the actual connection...
If there is a means on the device (a given port perhaps) to select whether the device will introduce the "cross over" itself, or not, then the cable you use doesn't matter. Just set the device to it's "other mode" if you don't get a link light or whatever...