The symmetry of the decad

A key component of the Minor Arcana is the series of integers from 1 to 10 -- the decad. Suppose we divide the decad into two parts (1-5 and 6-10) and think of the second half as the mirror-image of the first. That would mean 10 is the mirror-image of 1, 9 is the mirror-image of 2, and so on. We can depict this graphically thus.

I've marked the numbers for some basic properties. Odd numbers are black, and even numbers are red. Primes are in white boxes, and composite numbers are in yellow boxes. Squares and triangular numbers are marked with the appropriate polygons.

(The number 1, is of course, a special case where any of these properties is concerned. The ancients considered it prime but neither odd nor even; moderns consider it odd but neither prime nor composite. If the figurate numbers are defined by algebraic formulae, 1 is both square and triangular; if they are defined  by the geometrical figures for which they are named, it is neither. Since 1 is somewhat odd and somewhat prime but definitely not even or composite, I have coded it as an odd prime. My decision to mark it as not figurate was an admittedly ad hoc.)

Notice that each number's "mirror-image" is in fact its opposite in terms of all the properties shown on the graphic. Every odd number is paired with an even number; every prime is paired with a composite; every figurate number is paired with a non-figurate one.

I believe the decad is the only series of integers from 1 to n that has this property. I can say this with confidence because 2, as an even prime, must be paired with an odd composite. If it were not paired with 9, 9 would have to be paired with some other even prime -- but there is no other even prime.

Note the way 4 and 7 stand out on the diagram, as the only composite on the left and the only prime on the right. The Pythagoreans connected 4 with 10, and 7 with 1 -- and if we were to draw those connecting lines on our diagram, its symmetry would be preserved. Four was connected with ten because 1 + 2 + 3 + 4 = 10, the basis of the tetractys symbol. Seven was called "hidden unity" because, not counting the trivial sense in which 1 is a factor of everything, 1 and 7 are the only two numbers with neither factors nor multiples within the decad.