The vast majority of tartans are symmetrical. This means that when the pattern ends, the next repeat of the pattern will be in reverse.

If we can depict a tartan pattern numerically, for the sake of a simple demonstration, let's say a particular tartan runs 12345. If woven symmetrically, the tartan would look like:
12345432123454323123454321 etc...

This gives you two mirror points at each pivot (1 and 5) at which the tartan is the same to the left and right.

Now, an asymmetrical tartan does not reverse the pattern when it repeats. This means that your tartan would run like this:
1234512345123451234512345 etc...

Notice there are no mirror points.

Now, any tartan thread count can be woven symmetrically or assymetrically. But like I said, symmetry is the norm. There are some tartans that have been woven both ways, such as Berwick-upon-Tweed. But most tartans are intended to be woven one way or the other.

Just making up a tartan thread count, let's say we have a tartan that runs:
R40 G28 K4 G14
If this tartan is going to be woven symmetrically, that's all you'd need. Symmetry is always assumed. But if that tartan was intended to be assymetrical, an elipsis would be added at the end of the thread count, indicating that the pattern repeats without reversing.
R40 G28 K4 G14 ...

Most tartan design software defaults to symmetrical, but usually there is a setting for asymmetrical designs, as well.

Assymetrical tartans are in the minority, but some well known ones include Buchanan, Hunting Stewart, Campbell of Argyll, Old MacMillan, MacDonald Dress, and Malcolm.

There is no real historical signifigance behind assymetrical tartans that I am aware of. They are just a less common style of tartan.

Aye,
Matt