If the original vines were grafted plants, definitely remove root suckers and all growth originating below the graft union.

I think the most important thing re the growth on the top is to keep cutting back new whippy growth to encourage the formation of flowering wood. (The man who was responsible for many years for pruning the wisterias on the pergola at the nearby Botanical Garden was apparently nicknamed Edward Scissor-hands because of how much pruning he did!). As for what is growing on the uprights, I'd cut it back in - as opposed to completely removing it - in hopes of encouraging flowering wood there too.

I wouldn't worry too much about having several stems climbing the posts - as long as you are sure the stems are not root suckers if the plant was a grafted one!

Thank you for replying to me Woodyoak.

just so I can understand what you're saying about "new whippy growth", here's a photo of the top growth. Is everything (A + B) considered new, whippy growth? Should / Can I cut both kinds of growth to the woody stem?

Am i correct in thinking that flowers come off woody stems only? I think this might be our first flower bud!

A & B are new growth - but not at what I call the whippy stage :-) Since I have both a Chinese and a Japanese wisteria and the Chinese one will produce flowers in summer on growth that looks like that, I'm leery of cutting it back at that stage - even though my Japanese wisteria has never produced the secondary summer flowers that the Chinese one does :-) Leave those A&B for a few days and they will likely look more like this:

Re the flowerbud - the picture is a bit fuzzy so I can't tell for sure - but I suspect it may be just a fat foliage bud. Because the foliage on the Japanese wisteria emerges early, I remember how disappointing it was in the first year or two when fat buds turned out to be foliage, not flowers :-( My Japanese one started blooming at 5 years old - and there was no mistaking the flowerbuds! Here are a couple of views of the flowerbuds:

early in spring (old picture from first year of bloom - frost in the forecast, hence the protective cover!):

May 6 2017:

I think I definitely have a flower bud! Now since our primary concern is shade, I'm still a bit in the dark about that mass of growth at the top of the post. Everything I'm reading online says to choose one leader and cut it when it reaches the desired height. Instead, I have 6 woody branches reaching the top and I've allowed them to continue in different directions. Should I have pruned everything back but for one, and continued the "architecting" of the arbor with the subsequent branches/shoots of the main leader? There's was a definitive guide out there from Ohio State called "Growing Wisteria" that seems to have disappeared from the internets.

I only grow the wisterias as 'trees' so can't say for sure how best to grow them on a pergola. But I wouldn't worry too much about multiple stems up there. I think multiple stems would be desirable. I think the key issue would be cutting back the whippy growth coming off those main stems. Cutting those back will promote development of flowering wood. If you look at the picture I posted dated May 6 2017 you can see that there are a lot of stubby/shorter stems with lots of flower buds associated with them. Those shorter pieces are what was left after cutting back the whippy growth. On a pergola, cutting back all that whippy growth would also help keep the weight of the vine on the structure under control.

I find a lot of the detailed growing instructions not very useful/practical. If you understand a reasonable amount about how the vines grow, have a reasonably clear vision of what you want to achieve, pay attention to/observe closely what you vine is doing, and prune, prune, prune, you should have a good chance of success.