Based on the comments from Wendy Krieger below, I accept this observation from მამუკა ჯიბლაძე as an acceptable answer to the question
      
"product of all coordinates is 1 for E, -1 for O and 0 for A"
     
Comments from Wendy Krieger 11/06/2018:

    There is a geometric description here, which has to deal with 
    reflections.  A reflection in a mirror will produce a reversed image, 
    that is, convert 1 to -1.
        
    If you place two mirrors at right-angles, the image in the corner is a 
    double reflection, or -1 * -1 = +1.  So if you looked at yourself in 
    this arrangement, when you lift your right hand, the image lift its 
    right hand too.  In a single reflection, a right hand is reflected to 
    the image's left hand.
       
    The most likely source for this is the rectangular mirrors (the ones 
    like x=0, and y=0, and z=0), are flipping the image back and forward. 
    What it probably means in terms of your experiment is that A-X and A+X 
    somehow add to the same measure, but the difference from A is being 
    inverted.
        
    The whole of E8 is a eutactic star.  The two 24-cells are eutactic 
    stars, severally and jointly, and the thing you are looking at is one of 
    the 16chora inscribed in the 24-cells.
    
    The 24ch is the group of order 192, can be divided into three 4r's 
    (16ch) or lines at right angles.  These three can be labeled say N, E, 
    O.  The vertices of the form (1,1,1,1) etc, can be formed by an 'even' 
    sum of four N axies, or an odd sum of four O axies. even and odd here 
    means that the axies are labeled as +1 to -1, and an e/o number of 
    negatives are used.
         
    Likewise, (2,0,0,0) can be formed by an 'even' number of E or an 'odd' 
    number of O vectors.
       
    The signs then correspond to reflections in the N star, which inverts 
    exactly one axis (eg w,x,y,z -> -w,x,y,z) and to get to other points, 
    you have to reflect in the x, y, and/or z mirror too.
       
    The progression from O to N to E is a linear operation in odd 
    dimensions, but involves some sort of turn in even ones.  This is why 
    they can be treated symmetrically in 4 and 8 dimensions.

Added 20 Dec 2018:

For those familiar with at least the rudiments of biomolecular translation (the process by which messages transcribed from genes are translated into proteins), the following conceptual cheat-sheet may help explain why an affirmative answer to the "generalized Kronecker delta" question was so important to my research team.  In order to be able to deliver on the claims made in this cheat-sheet, we needed to be able to show that given Wendy's construction, we can define a "high" (+1), "low" (-1), and a midpoint between them (0).  And the affirmative answer provided to the question by  @მამუკაჯიბლაძე tells us that this is possible. 

    Conceptual Cheat-Sheet
    
    Our analysis deals solely with sets of dicodons and the sets of dipeptides
    they encode, NOT with individual codons and encoded amino acids, nor with 
    individual dicodons and encoded dipeptides.   
    
    Our analysis deals solely with the energetic properties of sets of dicodons,
    the affinity properties of sets of dipeptides (hydrophobicity), and the 
    synthetase affiliation properties of sets of dipeptides (Class I or Class 
    II).  No other properties of dicodons or dipeptides are relevant to the
    analysis.
    
    Our analysis identifies certain energetic symmetries in the energetic 
    patterns exhibited by our sets  of dicodons.
    
    Our analysis identifies certain affinity symmetries in the affinity patterns
    exhibited by our sets of dipeptides.
    
    Our analysis identifies certain affiliation symmetries in the affiliation 
    patterns exhibited by our sets of dipeptides.
    
    Our analysis identifies certain consistent symmetry relations between 
    energetic symmetries and affinity symmetries, and also between energetic 
    symmetries and affiliation symmetries.
    
    These symmetry relations hold for: i) both amino acids of dipeptides (AA1 
    and AA2 in a dipeptide AA1AA2); or ii) AA1 only; or iii) AA2 only; or iv) 
    neither.  And we interpret (i-iv) as suggesting that sets of dipeptides 
    assumed functionality in protein structure according to this rough 3-way 
    chronology:
    
    	Early onset of functionality: 	sets of dipeptides exhibiting symmetry
                                        relations to their dicodon sets for both 
                                        AA1And AA2
    	Late onset of functionality:	sets of dipeptides exhibiting symmetry
                                        relations to their dicodon sets for 
                                        neither AA1 nor AA2
    	Onset midway:			        sets of dipeptides exhibiting symmetry 
                                        relations to their dicodon set for AA1
                                        or AA2 but not botj
    
    Our rationale for this chronology is that early onset dipeptide sets assumed 
    functionality when mRNA energetics were still important in early translation
    systems, whereas late onset dipeptides assumed functionality in relatively 
    mature translation systems in which mRNA energetics were relatively less 
    important (primarily due to the advent of the water-tight ribosome.)
    
    This hypothetical chronology is fully falsifiable by determining if it makes
    correct predictions with respect to “early-late” pairs of SCOP protein 
    families within SCOP superfamilies (where protein family age is taken as the
    ranking assigned by GCA and his team.)