And Second Place goes to ALR, with honorable mention to Steve for being the first to guess the part's function.
Speaking of which, DADoES and I had a lengthy chat the other night about rebuilding my RCA-Whirlpool. He mentioned the filter, and what type it could be. From what he said, and my own investigation, the rubber dome which comprises one half of the filter, collapses during normal recirculation during agitation. This forces the lint to collect in the labyrinth created by the checkered surface of the rubber half and the maze on the hard plastic half. During drain, the flow reverses and the rubber dome expands, allowing the lint to be flushed away down the drain.
I was expecting the filter to be badly clogged, but it turned out to be quite clean when I opened it up. There were some flakes of hard water mineral depostis - probably got in there as I was trying to clean the outer tub. And also some hard water precipitate at the ends of the labyrinths, which was easily scrubbed away with hot water and a simple dish brush. The rubber is in very good shape; I think it must be a high grade, such as neoprene, to have survived so well for the past forty years, with no sign of cracking or deterioration. Not bad for something that flexes at least twice per wash cycle.
Next time I'll try to post something even more mysterious ;-)