How (the heck) does SOHO's SWAN camera image the entire 4π sr celestial sphere?

The answer is that it does it by scanning. There are two SWAN instruments, each of which sees an area of $5^circ times 5^circ$ at any one time: with a $25$-pixel sensor where each pixel is therefore one square degree.

One of these instruments looks at the northern hemisphere, and the other the southern hemisphere, and both of them are scanned by mechanisms which involve two rotating assemblies for each camera. Here is a rather grotty diagram of the north (up in the diagram) facing one I drew:

The camera is looking in the direction of the arrow, and the two rotating bits are represented by the two disks, and the second disk only rotates through the upper $pi$ angle.

However the camera actually sits down at the base of this assembly, looking upwards, with the light being directed down to it through two mirrors, one of which rotates with each assembly. I have not drawn these in as it got too hard to do the coordinate transformations.

To make life more interesting these mirrors are not plain: they are toroidal. The reason for this is that the mirrors do at least some (and I think probably all) of the focusing of the light, and using pairs of toroidal mirrors avoids astigmatism associated with spherically-ground mirrors. This is a neat trick.

And the final trick is that there is a cell full of hydrogen sitting in each optical path, which can be used to look for the Lyman-$alpha$ line in hydrogen. As mentioned in a comment, I thought I understood how this works, but I'm now not sure I do. Normally the hydrogen in the cells is molecular which means it does not absorb Ly-$alpha$. There are heaters in the cells which get them hot enough to dissociate the hydrogen in the cell, at which point it will absorb Ly-$alpha$. So I think they are using the difference between the state with the cell cold and that with it hot to infer how much Ly-$alpha$ they are seeing, as when it's hot they're essentially putting a Ly-$alpha$ absorber between the camera and the sky. But I am not completely sure about this.

References

• LATMOS are at least one of the groups involved in SWAN, and here is a page with a diagram of the instrument.
• NASA's SWAN page is less informative.
• Knight Optical have a useful description about toroidal mirrors and what they do.

Many of the references from the LATMOS pages seem to have rotted unfortunately as the project is quite old.