I have a .54 caliber Douglas barrel that is 1-1/8" x 36" straight octagon. It is stamped "G R DOUGLAS" and "54 CAL" on a flat near the breech end. The muzzle is stamped with the a "54", an "XX" (no circle), and either a "36" or "66". The latter either refers to the length or the twist, depending on what the first digit is. The X's are close together, but not perfectly aligned, indicating they were individually stamped.
The muzzle has a shallow recess for protection and the rifling is slightly tapered for about 3/8" instead of the more common radius crown.
I haven't mic'd for run out, but visually, I don't see any. The shoulder at the bottom of the thread section in the breech is a uniform thickness, so when it was milled and threaded, it was perfectly aligned with the bore.
This is the only Douglas premium barrel I have, so I don't know if the markings are typical or atypical. Hope the description is of some help when comparing to your barrel.