"Modes of the Major Scale"

People refer to the "modes of the major scale." I don't really get that -- it seems to me they're all modes of each other, not necessarily just of the major scale. That is, you can say that A-Aolian is a mode of C-Ionian, but isn't isn't it equally true that C-Ionian is a mode of A-Aolian?

My computer-science head wants to call C-Ionian, D-Dorian, E-Phrygian, F-Lydian, G-Mixolydian, A-Aolian, and B-Locrian something like a "mode class," which could be uniquely identified by any of those names (the "C-Ionian mode class", the "D-Dorian mode class," etc.) and would all be equivalent ways of referring to those same 7 scales.

OK, so each mode has a corresponding fret pattern (the pattern of frets that you play to produce that particular scale). We know that on the guitar you can change keys just by moving up or down the neck and playing the same fret pattern (G-major one fret up becomes G#-major) But that only works if the root you want is on the low-E string in the appropriate octave. What if it isn't? Knowing the above, can start with any of the notes in your scale, just playing the fret pattern that corresponds with the mode from that same mode class to get a "harmonic equivalent" (I just made that up).

That is, if you want to harmonize with C-major (or play a melody that's in the key of C-major) you could start on D and play the Dorian fret pattern, start on E and play the Phrygian, etc., whatever's convenient. So the point is you only actually need to learn those 7 fret patterns, and the system for determining which fret pattern to use from within the mode class (which is pretty easy, actually)

Not sure this is correct - I don't think I've seen anyone flat-out say it this way on a Web site. They talk about "the five patterns" that you need to play, but I think they're looking at it a different way -- as the major scales that can be played from a given position, I think, and you'd choose one of those five depending on where the root is relative to you hand. Maybe it works out the same way, dunno.