As a result of a recent discussion on stackexchange.com, I think I've found my mistake, and it's quite interesting. First of all, here is the picture I had of a typical mode:

The green lines are the electric field and the purple lines are the magnetic. You can see both fields to to zero on one of the faces. There are three choices for the blank face, so every unit cell gets three standing wave modes. Right?

The guy who posted the question on stackexchange (he goes by the awkward moniker of e-journal_clasica) started off by giving these formulas for the electric field of a standing wave:

My problem was to reconcile these formulas with the picture I had of the physical modes. I started off by noticing that his condition on E1, E2, and E3 was satisfiable infinitely many ways; but in particular, it was easily satisfiable for the choice of either E1, E2, or E3 =0. And weren't those my three mode configuarations?

Ah...but are they

*orthogonal*? That's when I noticed something very interesting, and I wonder if you can see what it is.

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