Selected additional examples to supplement my first answer:

Example 5

OP's original example, for completeness: 0,12,10,6,3,15,9,5,11,7,1,13,8,4,2,14 or $1\mapsto 12, 2\mapsto 10, 4\mapsto 3, 8\mapsto 11$ with $p=7$ and period $7M$ when $M$ is a power of two.

Example 6

The rule 0,1,4,5,2,3,6,7,10,11,14,15,8,9,12,13 or $1\mapsto 1, 2\leftrightarrow 4, 8\mapsto 10$ has $p=2$ and period 64 on a $32\times 32$-blocks grid, the product rather than the least common multiple of $p$ and the grid diameter. By a curious coincidence, the starting pattern I've used for all the other examples lives in an orbit half as long, so I've added a variant revealing the real period:

Example 7

With the rule 0,3,4,7,10,9,14,13,5,6,1,2,15,12,11,8 or $1\mapsto 3, 2\mapsto 4, 4\mapsto 10, 8\mapsto 5$ where $p=3$, with overall period $3M$ for $M$ a power or two, one can discern an antipodal copy of the starting pattern after half the period, but there's a wallpaper of other lit cells besides.

Example 8

With rule 0,2,5,7,10,8,15,13,6,4,3,1,12,14,9,11 or $1\mapsto 2, 2\mapsto 5, 4\mapsto 10, 8\mapsto 6$ (same period) it is even harder to discern a partial antipode. But since the rule includes summands amounting to $2\leftrightarrow 4$, lit SW and NE cells do still run northeast resp. southwest around the arena at light speed, and since $p=3$ is odd, they're at the antipodal position after half a period.

Example 9

In Rule 0,1,3,2,12,13,15,14,10,11,9,8,6,7,5,4 or $1\mapsto 1, 2\mapsto 3, 4\mapsto 12, 8\mapsto 10$, on the other hand, none of these summands are present. The single iterate $A$ involves only $T_{\W}$ and $T_{\N}$ and $T_{\E}$ terms. A small starting pattern branches out to the NW and N and NE for 48 iterations (when $M=32$), never reaching further than half the grid diameter towards the north, and then collapses in on itself over the remaining 16 iterations of the period. As in example 6, the period is $pM$, the product rather than the least common multiple.

Example 10

To illustrate the maximum $p=15$, let me close with one of my favorites: 0,1,2,3,5,4,7,6,12,13,14,15,9,8,11,10 or $1\mapsto 1, 2\mapsto 2, 4\mapsto 5, 8\mapsto 12$. It manages to replicate the input pattern already a quarter of the way around the grid towards the NW after a quarter period (when $M$ isn't too small for this to make sense), with additional copies popping into and out of existence at many other points. Only a quarter period is recorded to keep the file size reasonable; then the animation jumps back to the beginning.

When I first saw the expansion turning into cancellation and collapse, the phrase "How not to use a water gun" came to mind.