purpose of MOSFET terminal designation
In principle yes, they are somewhat symmetrical. In fact the original JFETs were symmetrical. Some JFET still has some symmetric behaviour, but I wouldn't bet on it unless it's clearly stated on the datasheet. In fact there is a different symbol for symmetrical JFET (the gate is on the center of the channel) while a common JFET has the gate arrow on the source.
However, no, you can't flip over a MOSFET and hope it will work correctly. It's actually a fabrication issue. If you look at the wikipedia drawing

it seems symmetrical. But if you look at how it's actually built (also from wiki)

(that is a DMOS, by the way, there are three or four other major ways to build them) you'll see that it is actually not so symmetrical (and that without looking into the dopant ratio of the various zones).
It also shows why the drain is always the heat sinked terminal, it's simply the biggest one and the nearest to the package.