A little more research of my own managed to turn up this absolutely lovely article: Ways of Coloring: Comparative Color Vision as a Case Study for Cognitive Science
Which contains comparative color spaces for humans, bees (also trichromats, but with different frequency response), goldfish, turtles (both of which are tetrachromats), and pigeons (suspected pentachromats). And it has an excellent statement of what the problem actually is:
It is important to realize that such an increase in chromatic dimensionality does
not mean that pigeons exhibit greater sensitivity to the monochromatic hues that
we see. For example, we should not suppose that since the hue discrimination of
the pigeon is best around 600nm, and since we see a 600nm stimulus as orange,
pigeons are better at discriminating spectral hues of orange than we are. Indeed, we
have reason to believe that such a mapping of our hue terms onto the pigeon would
be an error: [...] “Among other things, this result strongly emphasizes how
misleading it may be to use human hue designations to describe color vision in
non-human species.”
This point can be made even more forcefully, however, when it is a difference in
the dimensionality of color vision that we are considering. An increase in the dimensionality of color vision indicates a fundamentally different kind of color space. We are familiar with trichromatic color spaces such as our own, which require three independent axes for their specification, given either as receptor activation or as color channels. A tetrachromatic color space obviously requires four dimensions for its specification. It is thus an example of what can be called a color hyperspace.
The difference between a tetrachromatic and a trichromatic color space is therefore not like the difference between two trichromatic color spaces: The former two color spaces are incommensurable in a precise mathematical sense, for there is no way to map the kinds of distinctions available in four dimensions into the kinds of distinctions available in three dimensions without remainder. One might object that such incommensurability does not prevent one from “projecting” the higherdimensional space onto the lower; hence the difference in dimensionality simply means that the higher space contains more perceptual content than the lower. Such an interpretation, however, begs the fundamental question of how one is to choose to “project” the higher space onto the lower. Because the spaces are not isomorphic, there is no unique projection relation.
A common feature of all of the systems described is the production of a combined luminance channel from the raw n-dimensional cone cell inputs, as I suspected there would be, and n-1 oppositional color channels--in humans, these are the red-green and blue-yellow oppositions, which produce a two-dimensional neurological color space othogonal to the luminosity axis, corresponding to the classic color wheel, with fully saturated hues along the outer boundary and blue occurring across from yellow and red across from green. Saturation arises as the radial dimension--distance from the white-black axis--in a polar transformation of this oppositional color space.
In higher-dimensional color spaces, as determined by discrimination experiments on tetrachromatic and pentachromatic organisms, We still see the generation of oppositional color channels from retinal processing. How to generate these oppositional channels is not obvious; for example, in humans one opposition is between red and green, both of which are primary colors, but the other is between blue, a primary color, and yellow, a composite. Why that particular combination? It turns out, across different species, opponent channels are constructed to maximize decorrelation--in aother words, to remove redundant information caused by the overlapping response curves of different receptor types. Thus, the precise method of calculating color channels will be slightly different for each species, dependent on physical characteristics of the retinal cells, but they are all qualitatively the same kind of signal, and end up producing a a higher-dimensional hue-space orthogonal to the white-black luminosity axis.
Meanwhile, in any such neurological color space, there is only ever a single radial coordinate. Thus, we can say with some confidence that the extra dimensions introduced in higher-dimensional perceptual color spaces are not some extra sort of saturation or any kind of weird third thing, but are in fact additional dimensions of hue--and along with extra dimensions of hue, qualitatively different kinds of composite colors!
Our three dimensional human color space allows us to perceive two opponent channels, corresponding to 4 pure hues--red, yellow, green, and blue--and weighted binary combinations thereof--r+y (orange), y+g (chartreuse?), g+b (cyan), and b+r (purple), with one non-spectral hue. (Which, I suppose, means that the color wheel would still be a wheel even without the anomalous high-frequency response of human red cones!)
Meanwhile, a tetrachromatic system would have 3 opponent axes with 6 primary hues (r-g, y-b, and p-q), binary combinations of those hues producing secondary colors (r+y, r+b, r+p, r+q, g+y, g+b, g+p, g+q, y+p, y+q, b+p and b+q), and ternary combinations producing an entirely new kind of hue not found in the perceptual structure of trichromatic color space (r+y+p, r+y+q, r+b+p, r+b+q, etc.). Additionally, there is not merely one non-spectral intermediate color (purple) in the fully-saturated hue space, but 3--and in general, that number will correspond to however many pairs of non-spectrally-adjacent sensor types there are, which works out to the sequence of triangular numbers!