That's a decade of compromising images about to become significantly less anonymous...
A short explanation can be done using the Fourier transform. Blurs like Gaussian blur, and photographic camera blurs (with some simplifying assumptions) are convolutions; they apply a 'blur kernel' to each pixel of the image, which spreads energy from that pixel to all the neighboring pixels based on the shape of the kernel. Visualizing the outcome of a convolution is not straightforward for complex scenes, but here the Fourier transform helps.
When looked at in the frequency domain, the convolution operator turns into a multiplication operator; the spectrums of the image and the blur kernel are simply multiplied frequency-by-frequency. So you can directly see where information is being lost in the final image, by seeing what frequencies of the blur kernel are zero - at those frequencies, the output image has lost all original information.
Deconvolution techniques are all trying to restore the original image; in theory all you need to do is to take the blur kernel, and divide by its frequency spectrum to obtain the original. Assuming you have no noise, etc, in the process, this works fine, except where the blur kernel is zero - division by zero doesn't get you very far, and the information there is truly lost.
With camera handshake, the shape of the blur kernel tends to be a squiggle (the path of the camera motion), and the frequency spectrum is reasonably nicely behaved - there may be no zeros or just a few spots. So reversing the blur is possible, maybe with some additional interpolation to cover up the nulls. Out-of-focus blur (bokeh) is much harder, since it tends to be much more uniform and smooth, like a gaussian blur.
A gaussian blur turns out to have a gaussian frequency spectrum as well - that means the blur kernel has 0 frequency content past a cutoff point, and the wider the blur, the lower the frequency cutoff, and the more information is lost. So deconvolution can't really work directly; you can make assumptions about what was there before (priors), to guide the reconstruction. But at some point it's about as good as pasting a random face from the internet on the blurred head. The question is mostly about where that cutoff is - how much can your knowledge of 'this is a face' make up for the zeroed-out information? In practice, you're probably pretty safe if you've blurred the face to the point where no features remain. If you're really worried about it, throwing in some random noise, etc, makes the problem even more impossible.
So in short: We can probably do OK on camera shake and maybe out-of-focus bokeh. We can't recover from smooth uniform blurs like gaussian blurs.
This isn't a convolution, but the story's the same — the guy clearly thought that the 'swirl' was irreversible / lossy, but it isn't.
But I suppose a trapdoor blur -- a convolution that's difficult to reverse -- is possible. Maybe random noise is enough.