The trick, I realised, or problem if you look at it another way, is that he can't understand what I mean by "straight edges," so telling him to separate out the edge pieces first (as Special Agent Oso taught us to do back when we had the Disney Channel) is useless and frustrating to him. He looks at a piece and sees straight edges all over it, wherever it's not a definite curve.
So I stopped talking about edges and saw that he had already put together two sets of two pieces that were obvious matches. I asked him to find all the pieces of hand next (it's a rescue guy dangling from a helicopter reaching out a hand, drawn from the perspective of the rescuee, so the hand is huge and the rest of the man much smaller; such an odd picture to expect a child to figure out), as they're all the same colour. Then we did the helicopter, fitting larger pieces together as we we went along, and then we could easily fill in the last bits. He still tries to fit a corner piece right into the center, but I keep my mouth shut (mostly). Presumably at some point, just like with everything else, it will click and he'll understand what I was on about all along.
I think observing someone do a jigsaw must say a lot about how their mind works. I could probably work some finely crafted metaphor around it too. But I think I'll just mention that it's very nice to watch your almost-five-year-old work out a jigsaw for himself. (Especially while the toddler is napping and you have a cup of tea in hand.)