Just a quick note on the implications of the K numbers I talked about in my last post. (Edit - since deleted since there were a few issues with the approach - will try to come back and clarify this post at some point...)
In each primorial 0-30, 0-210, 0-2310 etc, there is a limited symmetrical pattern. Up to P#, K numbers (numbers that are either prime or don't have any factors or P or below) on the "way up" are composite, but K numbers on the "way down" (eg the remainder when a K number is subtracted from P#) are prime if they are below P^2. (If these remainders are over P^2 then they are either prime or they have a different set of composite factors to the K number.)
Not a very satisfying symmetry, granted, but it does explain why the assymmetrical prime patterns have such a symmetrical "feel". What we are actually seeing is a fractal, assymmetrical pattern gradually develop. But for every region P^2 to Pn^2 where Pn is the next prime after P, there is a symmetry with the region (Pn# - Pn^2) to (Pn# - P^2) with primes in the former being mirrored by Kpn numbers (numbers that don't have factors of Pn or below) in the latter.