The wiring diagram you linked to is too "dummed down" to seriously trust. If it is correct then the answer to your question is Yes, you can use one thermostat to both heat and cool by rewiring it.
To rewire it, I would swap the extension cord connectors, so that the male connector is on the right and you have two female connectors on the left. Plug the heater into one, and plug the cooler into the other.
But as I said before, this assumes that the wiring diagram is correct: namely that there is a SPDT relay of sorts in there, and that nothing else in the unit takes power from the AC lines. Some digital units run off of AC power, while others will take a battery. So figure things out before attempting this or you might blow up your thermostat.
There are some other important issues to keep in mind here: This thermostat, if rewired like this, will be always cooling or heating. Most thermostats have a sort of dead-band where it is doing nothing, right around the temperature it's set for. This is for an energy savings and to reduce "wear and tear" on the heater/cooler. Also, in most cases either the heater or cooler is not required so the dead-band acts to keep the unrequired part from turning on.
Also, there is a large chance that you don't actually need to have both a cooler and heater running. Normally when fermenting beer "at room temp" you only need to cool it. If your "room temp" is lower than your desired fermentation temp then you need a heater. If your "room temp" is fluctuating so much that you need both a cooler and heater then you should probably re-evaluate where you are fermenting.
And lastly, the wiring diagram implies that you can use a light-bulb as a heater. Don't do this. Exposing your brew to light will cause all sorts of problems: mainly "light struck" or "skunked" beer. Instead, use some sort of heater belt, like this one sold by Northern Brewer. Although light struck beer happens quickly with UV light, I have seen it happen with incandescent light too, it just takes longer. Also, a normal incandescent light won't last long in a fridge (I've had them die within a week or two) so you'll run the risk of your brew getting too cold when it fails.
It is only the A20 line; the origin of this hack was a bug in the 80286 processors that failed to set that line to 0 in real mode.
It also failed to set the others, but real mode programs did not care, so only that line got the gate.
Best Answer
A TEC is known in physics jargon as a "Peltier element" because it uses the Peltier effect. The peltier effect is semiconductor magic that cools one side of an array of p/n junctions while simultaneously heating the other side of the it. You should think of it more as a heat pump (thermodynamically) and not direct electric cooling.
A peltier is a curious beast. You can use it to heat something or cool something just by reversing current, with a proper control loop and a temperature sensor you can also keep something at a precise temperature. It also works in reverse, you can generate modest amounts of current into a low-z load by heating one side and cooling (or insulating from the heat) the other. There are a few consumer devices that use this to generate small amounts of power directly from a heatsource.
To answer your question. A CPU produces some amount of heat energy every second (Thermal Power), this heat needs to dissipate into the atmosphere at a rate equal to whatever the cpu is putting out. Cooling fins (heatsink) spreads that energy over a larger volume and more surface area for cooling and fans to create airflow increase the rate that heat escapes from the heatsink into the air. This is done to increase the amount of power a cpu can generate and still stay within operating temperatures. A TEC can be used to directly keep the CPU cool, but you need to use quite a bit of extra power to do so, and you have to keep the hot side of the TEC at a reasonable operating temperature so you will need an even bigger heatsink than you already have. If you let the TEC freerun as a peltier off the CPU heat you can divert a modest amount of heat as electric current to dissipate elsewhere, but this generally would be less effective than just using a larger heatsink.
In mission critical situations its easier to underclock and run the cpu conservatively than use active cooling, in hot systems or those driving them to extremese its more effective to use liquid cooling or just dunk the whole thing in non-conducting fluid. Otherwise CPU manufacturers are good at making CPU's that keep themselves cool with modest heatsinks, and a TEC is not a solution. The only situation I can think of where a peltier is a good design decision is if a constant or below room-temp operating temperature is a requirement.