Nyquist criterion for zero ISI in the frequency domain
The short answer is you will see that when you follow that symmetry, the resulting transform (the impulse response in time) will all have nulls spaced at T where T is the symbol duration which by definition is zero-ISI. Work through those transforms as well as ones without such symmetry and I believe it will become very clear to you what is going on. The simplest case to start is obvious: a rectangular function in one domain is a Sinc function in the other. If it is a rectangular function with width 1/T in frequency, it is a SInc in time with nulls at integer multiples of T.

Another very simple example is to then convolve the two rectangular fucntions in frequency to get a triangular fucntion which will have this "folded frequency" relationship. This is simply multiplying in time, so although we changed the Sinc response, we have not changed the location of the nulls.