You need symmetry.
If you shift the parallelogram you violate symmetry.
A standing wave is not a scalar but a vector creating a fixed, symmetric pattern in a room.
Antinodes are always in the corner, pressure flows to boundaries.
"I'm also confused by how the length/width of the sides of the standing wave parallelogram relate to the length/width of the sides of the room. In Rayleigh's equation we seem to be only plugging in numbers related to the room dimensions."
I'm not sure I understand your question.
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