I have a great 5 yr. old honeysuckle growing on the south side of a small gazebo. Of course it all leans towards the south (sun) and is now very weird looking, creating a lopsided gazebo top (see attached photo - it's the brown bushy stuff on the right of the gazebo.) I've tried tying it back towards the other side, but those branches died. Is there anything I can do? I also planted some clematis on the opposite side, but it doesn't bloom/spread as much; not enough to offset the lush honeysuckle growth.
Ramsey County Minnesota honeysuckle
Well, it's hard to argue with the sun, and a plant's basic nature. You have rightly pointed out that the plant is searching for sunlight in order to grow as healthily as it has.
You can certainly prune this plant in order to shape it a bit, and to thin it out a bit. Rejuvenation pruning is commonly done on honeysuckle vines. Thinning it may be more important than shaping it, as good air and light flow through the plant helps keep diseases at bay, and reduces places for insect to hide.
Here is a link that should help you with both of your vines: