1. Forget anything you know about symmetry classes; they do things differently there.
2. A pattern is not a tartan unless you can fold it diagonally so that the colours match. More in particular...
3. There are two kinds of tartans, symmetric ones and asymmetric ones. Almost all tartans are symmetric, which means that they have two different bands - horizontal or vertical - around which the pattern can be folded.
4. An asymmetric tartan has no horizontal or vertical bands around which it can be folded. The following is an asymmetric tartan:
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