Doesn't a lens also help flip the image back to the scene orientation?

Lenses allow more light to pass through, so you can capture a scene with less exposure(time).

I don't think a lens does any flipping operation. Correct me if I'm wrong...

It looks like the image is still flipped (notice that the scene object 1, which starts on the left, ends up on the right as object 2). The reason we don't see everything upside down after the lens in our eye flips it is because the brain corrects for this. https://www.scientificamerican.com/article/experiments-with-goggles/