The 5608 is a simply a medium Mu, duo-triode with a common cathode (pin 4).
The 5608 was replaced by the 6J6:
https://frank.pocnet.net/sheets/093/6/6J6.pdfOne could use it as a dual medium mu audio triode. With the strap the filament will be at the same potential as the cathode.
I saw this once in a boiler control. it was the first AND gate so to speak.
One of the grids sensed pilot light heat via a thermocouple, the other grid went to an "on" or
activate switch. Both plates went to the AC supply.
The cathode went to an AC relay that when pulled-in, opened the main burner gas valve.
Phil - AC0OB