Thanks! I have to think about the $SO(4)$ action in a quiet room for a few minutes, but something like "the $n^2$-dimensional space is an irreducible representation of a larger set of symmetries" is totally the kind of answer I was hoping for.

The rows correspond to atomic numbers 1-2 (length=2), 3-10 (length=8), 11-18 (length=8), 19-36 (length=18), 37-54 (length=18), 55-86 (length=32), 87-118 (length=32). The two sections of length 14 that you're talking about are subsections of the two rows of length 32 (look at the atomic numbers and see).

The 14 elements are just a section of that row, the entire row is $18 + 14 = 32 = 16 \times 2$ elements. Each section of $14 = 7 \times 2$ elements corresponds to $V_6 \otimes W$, and the $7$ gets added on to $1 + 3 + 5$ to give you your next perfect square.