As previous mentioned, the eyes of a jumping spider act like mirror balls.
The eyes themselves are not quite a full hemisphere, but still they reflect what may be a surprisingly wide angle -- in fact more than a full hemisphere!
Here's a quick summary:

Notice in particular that the spider is positioned at the edge of a balsa sheet, and the sheet itself is clearly reflected in the spider's eyes.
Meanwhile, the face of your earnest investigator, standing patiently with his chin on the hot shoe of the camera, occupies only a small portion of the reflected field.
It may be amusing to contrast the actual reflection shown above with the artist's conception recently posted in another thread.
On reflection (pardon the pun...), what comes to mind is
CAUTION: Objects In Mirror Are Closer Than They Appear!



On a more serious note, I also shot the critter with a variety of illumination setups , including the classic pingpong ball, a large diffuser sitting over the subject, the same diffuser at the end of the objective, a bit farther back, on a tripod farther yet, and finally without the diffuser as far back as the cord would reach. I also used a couple of different objectives and played around with masking one objective when there was a distracting reflection from its shiny chrome barrel.
Quickly summarizing those results:

I won't bother to show the details of all the lighting setup at this point. But I would like to illustrate one particular setup that seems especially effective:

The key aspects of this setup are that
a) it does not include a wrap-around diffuser,
b) the diffuser is positioned behind the tip of the objective, and
c) the diffuser is positioned as close as possible to the subject, consistent with (b).
If you read carefully the names of the images as shown in the title bars, you may notice that the last four images say "retouched" while the first four do not.
That distinction is important.
What happened in these tests is that when the illumination was quite diffuse, the stacking process produced a good result by itself, but when the illumination was not diffuse, the stacking process produced very bad artifacts that needed heavy retouching. In each of the images that says "retouched", the stacking result had such serious artifacts around the highlights that they were completely unacceptable. In those cases, I essentially replaced the eyes with the most attractive single-frame rendition. That resulted in a sharp reflection, but at the cost of losing surface detail.
For myself, the main conclusion is that for this type of subject I'll use a single flash with a fairly large diffuser, placed as close to the subject as I can without directly illuminating the front of the lens.
I hope this is interesting and helpful. More later, time permitting...
--Rik
PS. Yes, the first panel was shot with some unusual illumination. To get a really bright reflection, I arranged to illuminate the subject by ambient room light while using a bright flash to illuminate the front of the objective and my own face, but not the subject. The reason you don't see a flash cable plugged into the camera is because it's plugged into a different control box that allows for long shutter with EFSC for the ambient exposure, combined with flash in the middle of the long exposure. Some people may be interested in that aspect, but that's for another day.
Edit: to add references