How to reverse frequency spectrum of a 2D image?
I think that you should see it as doing it in one dimension and then the other, meaning you'd invert the values of every other pixel in every row, then once it's done do it again for every column, which I suppose means you can do it directly in 2D by inverting pixels in a checkered pattern like this:

1 -1
-1 1

As for your FFT/IFFT question yes it should be lossless, save for small computation errors, and I'm not sure but it might only work with even sizes so the Nyquist and DC components can switch places. However in both cases you'll have to deal with negative pixel values, so that kind of limits you to your own algorithms, unless you do $f(v) = 0.5+0.5v$ on all your values and then undo it.