Hi, I don't really know what you mean by double clothesline poles but it sounds like the setup my grandparents had years ago to grow grapes on. Both sets of grandparents were Greek immigrants who grew their own grapes. It is lovely to sit under a grape arbor. I don't remember this being a very elaborate structure - I seem to recall some kind of wire fencing covering the top and the grapes grew over it and the spaces in the wire fence were large enough for the grapes to hang down.
If I have the right picture in my mind the clothesline poles are like great upside down squarish U's some distance apart. I would use some kind of lattice on the ends, frame in the top with wood and staple my wire fencing over that.
If you want something prettier you could frame in the top with wood and then put cross pieces like any other arbor.
If you live in a city with a Greek or Italian neighborhood you can snoop around and see how they do it.
Yes, I definitely need a grape arbor, too!

Thanks for the wire fencing suggestion. I may need to put something for support half way in between the two uprights. It is a long distance.

I think clothesline poles are usually shaped like a T. The double poles have two upright posts, one about one-third of the way in from both end. So, I guess they are more like a Pi symbol.