Yes. The difference between Symmetrical, and asymmetrical is this (as I understand it)...

In a symmetrical tartan, the pattern of the warp is mirrored from the beginning to the end of the sett.

I.E. bk2 w4 r12 b2 r12 w4 bk2.

In an asymmetrical tartan, the sett is not mirrored.

I.E. bk2 w4 r12 b2, at which point it repeats the sett bk2 w4 r12 b2, and so on, instead of mirroring.

Did I get that right?

In any case, the weft will be exactly the same as the warp, no matter what the warp does.