"Spiral filaments" refers to construction inside the 12AX7. It's a technique for reducing hum when the filaments are AC powered. Not the case here, so it's irrelevant. About your filament wiring, It will make no difference whether it's twisted or how tightly. That's only an issue for AC heating.
Can you tell whether the hum is 60Hz or 120Hz? Also, what is the third wire ground from the power inlet connected to?
Jack
I think 60Hz but that is a guess. Here's some pics of the wiring. If you are referring to the a/c filter earth ground, it has two wires connected to it. The purple transformer shield wire and then the other wire connects on the power board as ground.

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