Excellent visual!!!

Here's another way of thinking about it. The pivots represent lines along which you could fold the tartan and have the two halves be exact mirror images of one another.

For the Black Watch, either the "B" or the "C" can be a pivot

ABACA B ACABA or ACABA C ABACA

The tartan is symmetrical because the sequence of letters is the same going either direction from the pivot. The fact that the "A" cannot be a pivot is immaterial.

A tartan with an ABCD pattern can't be symmetrical, because, no matter how many times you repeat the sequence, it's impossible to find a line across which the sett is a _mirror image_:

ABCDABC D ABCD nope; ABCDAB C DABCD nope; ABCDA B CDABCD nope; ABCD A BCDABCD nope

Cheers!

Barb