[DavidK]

If the "spider" is the paper swirl with 3 "tabs", two of them were torn when I got it.  Based on photos I've seen, I considered it part (tip) of the "cone".  There's a small center bolt that connect the "cone" to the floating "coil" inside the electro-magnet coil.

I'd like to use the audio transformer and field coil with the repaired cone but I don't know exactly how the main chassis gets both power and signal to it via the 3-pin connector.  Then I could rig up a "test connector" and "drive" the speaker separate from the chassis.  I'll squint harder at the schematic.  

I may end up back at my initial decision to just sell what I have and put in a modern mid-range (and tweeter?) but I wanted to check my options for reusing more of the existing stuff. Goodwill, Salvation Army, and Garage Sales are my main "suppliers".