With a lot of help from george2079 and Jack LaVigne, I have arrived at a solution to my issue.   

 1. Part of my problem was the syntax for my pure functions  
 2. Additionally, GatherBy does not order the subgroups in a "sorted" way.  
 3. I couldn't "distribute" a pure function Sort over a list of lists.  
Given these considerations, something like   
    GatherBy[Sort[rings[3],Lmag@Coefficient[#1, {q1, q2, q3}] < Lmag@Coefficient[#2, {q1, q2, q3}] &], Lmag@Coefficient[#, {q1, q2, q3}] &];  
Will give the proper subsets ordered by their Norms. First, you must Sort the list rings[3] by magnitude, and then GatherBy to separate the subsets. For some reason, ***if you don't sort them first, GatherBy will arrange the subsets in order of the first appearance of each equivalence class!*** *This behavior is unusual, because GatherBy does not work this way for the Numerical case!*  
Now, I don't have mma v10 which includes a similar command GroupBy (I actually can't tell how it differs from GatherBy) so there is a chance, I suppose, that function might accomplish both goals simultaneously.   
In the numeric case, I could then simply "distribute" a Sort over these lists of lists to sort by the ArcTan function. Perhaps if I were more adept at pure functions, this could still be done in a single step, I'm not sure. Nonetheless, it can be done by simply sorting each sublist individually, which is fine for my purpose.   
**With that, should anyone come along and care to see it, here is a fully functional solution to my original post**  

    Block[{order = 4, rings, splitrings, stars, q1, q2, q3}, rings[0] = {0}; stars[0, 0] = rings[0]; rings[1] = {q1, q2, q3}; Do[rings[r] = Complement[
    Flatten[Table[rings[r - 1][[s]] + (-1)^(r - 1) Boole[\[Sigma] ==1] q1 +(-1)^(r - 1) Boole[\[Sigma] ==2] q2 + (-1)^(r - 1) Boole[\[Sigma] == 3] q3, {s, 1,Length[rings[r - 1]]}, {\[Sigma], 1, 3}], 1], 
    rings[r - 2]], {r, 2, order}]; Do[splitrings[r] = GatherBy[   Sort[rings[r], Lmag@Coefficient[#1, {q1, q2, q3}] < 
       Lmag@Coefficient[#2, {q1, q2, q3}] &], 
    Lmag@Coefficient[#, {q1, q2, q3}] &], {r, 1, order}]; Do[stars[r, s] =    Sort[splitrings[r][[s]], L\[CurlyPhi]@Coefficient[#1, {q1, q2, q3}] < 
      L\[CurlyPhi]@Coefficient[#2, {q1, q2, q3}] &], {r, 1, order}, {s, 1, Length@splitrings[r]}]]