Peugeot Mixte

which I wouldn't expect on this bike. Along with the large fork rake, typical for a French bike, it looks to me like the wheel touches the road in front of where a line through the steering tube hits the road. If I'm seeing right, that would make the bike unstable. Can you ride with no hands? If it's unstable, ride the Benotto when you want to go fast. It's still a lovely bike to ride around the neighborhood.

Skipdiver is probably guessing that it's been in a head-on crash, that could cause the steep head angle. Then it would have had a new fork, probably.

you know, I don't think it's damaged - the head tube looks parallel to the seat tube, once you get your head around the angle of the photo. I've just bought a Peugeot mixte that's the same in terms of construction, so it seems unlikely that two bikes in different places have suffered identical damage and have been repaired identically.

Perhaps the bars aren't original, though? - that bend is an odd angle!

I've actually been wondering about the steep head angle myself, after adjusting the saddle height and moving it a bit forward handling improved a lot but it did feel very twitchy when I first rode it. I can now ride with no hands and shift the gears on the impaling stem shifters without getting (too) wobbly but it's definitely a peculiar ride.

Upon further inspection of the frame, I now do believe that the head tube has been bent. Does anyone know if it's possible to bend it back?

It's possible to bend a steel frame. Every time it is bent, in collision or repair, it's going to be a bit more brittle or weaker.

If you are able to ride with no hands, maybe you can live with it as it is. Depends on you and what you want to use it for.

If you want to fix it, and feel adventuresome, you could remove the fork, put a steel bar or pipe in the head tube, and try to bend it straight. If you value the bike more, and can find a frame builder or a bike shop that has experience with that kind of thing, take it to them.

Good luck with it, whichever way you go.