John Woo hops right into the story with "Face/Off". The first scene we are presented shows us not only these two rivals, the terrorist Troy and cop Archer. It also shows one of the most pivotal moment's in Archer's life, which forever after he becomes a changed person: Castor Troy kills Archer's youngest son in an attempt to assassinate Archer. Their whole rivalry is because of this moment. Archer becomes a changed man. He spends the next couple years of his life trying to finally catch Troy. This becomes the crux of the story. Everything falls back upon this. It is only when this is finally resolved at the end once Castor Troy is finally killed for good that the family, along with Archer, can finally move on as they accept a new adopted child to the family.
The story is very straight forward temporally. It is after all a major blockbuster type film, which is conscious of this fact. The film doesn't take itself too seriously and isn't afraid to go into huge crazy spectacle driven fight scenes. However, since the film does follow heavily both Troy and Archer, there is a fair amount of crosscutting and some simultaneous action. For example, we see Archer break out of the prison that he is put in as Troy after he swapped faces with him, but we also see what Troy is doing during this time to stop him from his office in the CIA. The film itself is
surprisingly clean. All the characters' arcs are complete by the ending of the film. Castor Troy is now finally dead, Archer's family is finally at rest, Archer chooses to let go of the loss of his son by choosing not to get the bullet wound scar reapplied to his chest, and they adopt a new son. It is a traditional and intentional Hollywood ending.
The story follows Archer first and then Castor Troy second. We are not restricted to one character's knowledge. We actually get a substantial amount of dramatic irony throughout the film, since we know what both of these two unstoppable forces are doing and how they are plotting against each other or walking into each other's traps. It is a live action game of Spy v. Spy in the form of cop v. terrorist. We do get an idea of the characters' mental states but only through performance and some flashbacks.
Do not be confused, this is a Hollywood film. Or it is a Hong Kong/John Woo style Hollywood film.
After all, they do swap faces in the movie.