Yes, 360's are different, exposure varies a lot depending more on the lighting angle and lighting distance, plus the reflectivity of the subject and this is both inevitable and unimportant.
I flirted with 360's a couple of years ago. Firstly we had a couple of products shot by one of your competitors, it worked as proof of concept but we were very unhappy with the quality, partly because of poor internal communication, the person who actually did the jobs obviously hadn't been briefed properly and the products were shot at the wrong angles, they were also flooded with light and could have looked much better. Again, horses for courses, their kind of approach would have worked perfectly for subjects such as shoes but weren't good enough for our specific needs.
We then looked at buying in off-the shelf solutions, which were not only very expensive but also produced overexposed and overprocessed images, a very easy way of doing it but not good enough.
So, we made up our own kit. We set up the subject so that it was lit perfectly for the very first shot (the one seen by the customer before the rotation starts) and used pretty hard lighting to add drama and to make the subjects look as interesting as possible. Ours were 36 shot animations, move the turntable by 10 degrees for the next shot and the lighting became wrong, another 10 degrees for the next and it became hopeless, but none of it mattered at all because the "wrong" lighting just made the product look more interesting. The colours (using flash) hardly shifted at all but they looked as if they did, again it didn't matter, it just added to the visual effect. The shots below effectively have little colour but some other products were fairly colourful.
The hard light, set up to make the softbox have a bright white interior created over-bright diffused specular highlights on the fabric but again this added to the effect instead of detracting from it.
Each of our 360's was a one-off, lit specifically for the qualities of that particular product and we didn't overexpose or overprocess the images to get a white background - we cleaned up the background for each individual image instead, which gives much better results but which cost about an hour of labour per product - which was well worth it for us but which obviously isn't practicable for everyone.
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Unfortunately this is no longer being done so I can't show you any finished animations but here are the individual images for one of them. It's a softbox fitted to a flash head, which is fitted to a spiggot for support, later removed in PP. If the pic is big enough you'll see that at various points we removed and replaced both the inner and outer diffuser, providing more info to the customer.