Jiro is correct!

Twins and antipodes, solidus and its converse
Refers to the nature of the scars, they are "twins" because they're pretty much the same and were received at the same time. They are "antipodes" because, as "solidus and its converse" says, Seifer's squall is shaped like a solidus ( / ), and Squall's is mirrored ( \ ).

Well known the bridges and fronts we traverse
The scars both cross Seifer's and Squall's nosebridges and fronts. Well known because Squall and Seifer are such prominent characters.

Painful was our beginning, but we welcomed you
As Jiro says, the scars were painful to receive. "We welcomed you" refers to the fact that it's one of the first things you see happening in the game.

You won't see the end of us, though we each have two
The scars remain for the entirety of the game, but they do each have two ends (pertaining to their shape).

Jiro is up!