They basically bought an 80%+ completed bike from Triumph and added cosmetic/styling bits - or that was my understanding.
As for the name, nobody in the mainstream buying market today remembers Trident except in regards to chewing gum and maybe in conjunction with nuclear missiles. If you check the Wikipedia article on the Rocket III, you'll see that they focus grouped the name and what came out of the focus groups was, "Don't call it a Trident, call it the Rocket."
Speaking of Wikipedia, I see they also have a page for the BSA Rocket 3 / Triumph Trident.
There's some really interesting tidbits there and it seems we both have it wrong.
BSA made all the engines, not Triumph.
The BSA had a twin cradle tube frame, while the Triumph had a single down tube design.
The BSA engine was canted forward in the frame, with the cylinders appearing slanted.
The Triumph engine was mounted flat, with the cylinders appearing vertical.
The Triumph engine remained mounted as such until 1975.
Interesting history, that's for sure.