Again there are two ancestors per state, giving squares of 4 and the
minimal element count of 8 for N. This time 
and 
are
idempotents with eigenvalues 0 and 1 but they do not commute. N
is nevertheless ``idempotent'' according to 
; its element sum
is 8 with powers growing at the minimal rate.
The difference is that Rule 12 is reversible, via a left shift, whereas Rule 6 is not reversible at all, each different value of a boundary cell resulting in a completely different ancestor (or no ancestor at all if the left and right boundaries conflict). Rule 12 is indifferent to its right boundary, but the ancestor is uniquely determined by the left boundary. Yet both rules are surjective.
Both these rules have a pair matrix with eight 1's, the minimal number
possible; uniformly distributed throughout the rows and columns for
Rule 6, but not for Rule 12. The difference is crucial, and is related
to the observation that rules of the Fredkin type are all constructed
from groups of variables, some of which are superfluous. Right shift
does not depend upon the right cell, so the matrices 
(and their
tensor squares) have constant rows, bunching the 1's in a way that does
not occur for Rule 6.