I was going for nature-philosophical rather than magical, honestly!
The foundation of the system is the idea that nature can be "parametrized" in terms of two quantities, hardness and wetness. The elemental values of hardness are threefold, namely fleet (meaning intangible), soft, and hard; and those of wetness are twofold, namely sere (meaning dry) and wet. The six elements are the substances defined by the pairings one gets from those. Thus, FIRE is the "substance" that is fleet and sere, AIR is fleet and wet, WATER is soft and wet, ICE is hard and wet, ROCK is hard and sere, and ASH is soft and sere. Clearly, the idea of hardness is quite similar to that of the physical phases of matter (gaseous, liquid, solid), and I did try to define one of the axes as temperature originally, but eventually came to the conclusion that that wasn't the way to go here: Simply put, air and rock are both room temperature, as far as human(oid) experience is concerned, which makes the connection between heat and phase too abstract to work with.
Thus, the circumstance that the fleet-sere element is hot and the hard-wet element cold is regarded as little more than incidental, in this version - unintuitive as that is to us scientifically literate humans.
Also, maybe the "+" symbol was a bad choice on my part. By "FIRE + ROCK", I don't so much mean "what do you get if you put one part fire and one part rock in a cocktail shaker", but rather "what lies between fire and rock" - and that has to be ASH by virtue of looking at the elements in terms of their hardness and wetness values.
That's why I'm thinking that, for example, Snow lies between WATER and ICE - it's wet, and it's halfway between soft and hard. The equalities in line 3) follow directly as well, because they all point to the same intermediate substance - one which is soft and moist. Earth fits that description, and, as I said, is just too perfect a candidate for the very centrepoint of the whole scheme to not use it. Plus, if one does allow a bit of mysticism to seep in now, coming up with explanations for each of the combinations is quite straightforward, too: FIRE + ICE = Earth because blah blah furnace of creation blah blah. AIR + ROCK = Earth because earth is spatially the layer between air (atmosphere) and rock (crust). And WATER + ASH = Earth because, well, for this one we can actually pretty much use the cocktail shaker.
Similarly, ROCK + ICE really ought to be something hard but moist, for which nothing comes to mind (well, perhaps "frozen clay", but that's just too lame to consider, I'm thinking). Slightly less literally, something hard and with some of the aspects of rock and some of those of ice. As I said, I'd be tempted to put metal there, if I wanted to have it in the scheme at all, because it does sort of fit that description.
So, thank you kindly for your suggestions, but they don't quite follow my overall train of thought here. Sorry for not having explained better in the first place (I did explain most of the above at some point in some thread, but even I can't remember which one that was, so that's not much of an excuse.)
Soon the Cold One took flight, yielded Goddess and field to the victor: The Lord of the Light.