These 
gold-coated hexagonal segments form @NASAWebb’s super-light, super-strong primary mirror — and allow it to fold up and fit in the nose cone of a rocket! 
It’ll help us see the first stars and galaxies of the early universe, and so much more: https://go.nasa.gov/31XI7Ic pic.twitter.com/4ldEu1oYlR
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It's only three folding sections, I see from your link. I had a sort of nightmarish vision of all the hexagonal pieces being stacked and folded and having to twist and move all around to get in place. Engineers do better designs that my nightmarish visions do.
You are correct - the side wings (3 mirrors each) fold backwards like a drop-leaf table. And the secondary mirror also folds. All 18 mirrors do have their own actuators though, which allow the mirrors to be fine-tuned w/ 6 degrees of freedom + curvature.