: I am making my ergometer and don't know how to find a part.
: It is the 'twist-joint' that I am looking for..

I don't see why a twist or swivel joint would be necessary, as there is not enough twisting motion involved that the rope and bungee couldn't easily absorb it. You can buy plastic clip hooks for bungee ends at marine suppliers like West Marine. Tie or splice a loop in the rope, hook it to the bungee and you're done.