Short answer: Can you? Yes*. Should you? Probably not.
It's not so much a question of whether it's physically possible, but more of a rule of thumb to prevent the two systems from accidentally interfering with each other.
*If, as you say, the units are separate from each other, and have separate thermostats, then they should be completely capable of being run simultaneously. In my house, I can certainly switch on both units at the same time. Theoretically, if you had the cooler set to kick on above a certain temperature, and the furnace set to kick on below a LOWER temperature, then only one would kick on at any given time.
However, I think it's more of a safeguard to not have them switched on simultaneously. Traditionally, the heating thermostat is in a fairly central location, which also tends to be where the main swamp cooler vent is best located. This means that the temperature at the thermostat is going to be influenced by the direct stream of cold air, which will get warmer as it distributes through the house. So it's far more likely to hit it's "switch-on" temperature and turn the heat on, when you in fact wouldn't want that.
And of course if you accidentally set the heat too high (above the swamp cooler thermostat), it would heat the house to the point that the cooler would kick on and you'd end up in a very expensive feedback loop as your house is simultaneously trying to both lower and raise its temperature.
The other thing is, in cold weather, the swamp cooler should be winterized and covered, since it's a direct vent to external air and would lose a large amount of warm air generated by the heating system. If you're heating with the cooler uncovered, you'll be losing a lot of that heat out the roof.
Edit: Since most swamp coolers blast a the air into a single vent, the cooling isn't very evenly distributed throughout the house. Ed Beal's solution of having the air handler fan would be a good way to circulate the air to get that cool air a little more even. I personally ducted my swamp cooler into the individual rooms to get better distribution.