The film is chock full of examples of Fridge Brilliance, many of them being plot points and scenes that only make sense upon a second viewing when you know the twist:
When Jack is beating himself up in his boss's room, he says that it reminds him of his first fight with Tyler, but he doesn't know why. Later, we realize that it is because Tyler is a split personality of the narrator's, so when they were first fighting, he was beating himself up.
During the one scene in the film in which the narrator behaves exactly like Tyler, when he's threatening his boss after the latter finds the list of fight club rules in the copy machine, he says in voiceover (presumably literally): "Tyler's words were coming out of my mouth."
In the scene on the bus when Tyler talks about the underwear ad ("is this what a man's supposed to look like?") a guy avoids bumping into the narrator, but bumps into Tyler (of course he was just walking trough an empty space, but since the narrator is seeing Tyler there, he sees the guy bumping into Tyler).
When the narrator and Tyler walk up to the convenience store, Tyler takes a gun out of the narrator's backpack and the narrator asks Tyler "Is that a gun? Please tell me that's not a gun." Despite the fact that it's his own backpack.
Near the end of the movie, while the narrator is flying all over the country looking for Tyler, he informs us in voiceover: "I was living in a state of perpetual deja vu. Everywhere I went I felt like I'd already been there."
During the montage early in the film in which the narrator explains how his job requires him to fly around the country, he asks at one point "If you wake up in a different place, at a different time, could you wake up as a different person?" and the camera follows Tyler.
In one scene, the narrator returns home from work early and starts cleaning a blood stain on his trousers while Tyler and Marla noisily have sex upstairs. When the police detective calls and the narrator answers the phone, the sounds of lovemaking instantly stop.
On the airplane, the narrator remarks how he and Tyler have the exact same briefcase. Tyler then opens his, but we never see the contents of the narrator's.
When the narrator and Tyler enter Lou's Tavern for the first fight club meeting in the basement, the guy standing by the front door nods only to the narrator.
When the narrator and Tyler first start bashing car bumpers with baseball bats, Tyler hits the car first, but the alarm is triggered only after the narrator hits.
When the airport employee "lends" Tyler the car, he pulls up, gets out, and says "Don't worry, Mr. Durden. Airport parking, long-term," but looks at the narrator, not Tyler. The narrator and Tyler then both enter through the driver's door and after the car flips over and falls into the ditch, Tyler crawls out of the passenger side, and pulls the narrator out of the driver's side.
During the scene in the car when the narrator and Tyler are arguing, the two Space Monkeys in the backseat look uneasily at each other at one point, because the narrator/Tyler is driving the car and is talking to himself.
After the car accident scene, the narrator lies in bed recovering with a large bruise on his face while Tyler talks, then Tyler gets up and leaves with a briefcase in hand. The next day when the narrator wakes up, the bruise is gone. Some may view this as a continuity error, while others may treat it as an indication that several days, perhaps a week or even more time, has passed, during which the narrator was not really asleep but was flying around the country as Tyler setting up the new fight club chapters.
Tyler gives the whole go-out-and-lose-a-fight assignment so that the narrator wouldn't realize they were the same person. Tyler needed to cover up the injuries and no one would have agreed to fight the narrator after seeing Lou's brutal beating. If the narrator didn't get beat-up, he was bound to wonder why he was injured when it was Tyler who had been in the fight.
You may not notice it on a first viewing but when Marla is giving her odd speech about condoms and bridesmaids' dresses she's actually talking about herself: all her life (perhaps since grade school...?) she's felt like a beautiful object that people use once or twice and then heartlessly thrust aside. This is why she gets even madder than you might expect when she thinks Tyler is trying to pull a one night stand on her.
It makes the narrator's/Tyler's response, "Well then it suits you" even harsher than at first glance. He's essentially confirming that he only sees Marla as an object to be used and thrown away. No wonder she storms off afterwards.
The narrator first discovers Marla in his house when she walks into his kitchen and starts acting all flirty, saying "I can hardly believe anything about last night." The narrator then asks "What are you doing in my house?" and she angrily storms out, because it was Marla and the narrator, whom she thinks is Tyler, who had sex the night before. When Tyler then comes in and starts to explain how he and Marla met, the narrator tells us "I already knew the story before he told it." The movie then flashes back to show Tyler walking up to Marla's apartment and knocking on the door. She answers it, eagerly pulls him inside, and says "You got here fast!" because it was really the narrator who showed up and she had been talking with him on the phone a few minutes earlier.
When the narrator asks Marla what she gets out of her relationship with Tyler, she dodges the question and asks "What do you get out of it?" The narrator thinks she is asking about his relationship with Tyler, but she is actually asking about the narrator's relationship with her. The narrator then starts to investigate the sounds of Tyler making a ruckus in the basement, but Marla says she doesn't hear anything.
When the narrator and the Space Monkeys are watching the news report showing the defaced office building, the narrator asks them "What did you guys do?" and they start to laugh, as if he had a made a joke about not knowing what was going on.
You'll also notice that when the narrator walks into the house during this scene, he is holding the various folders with the building information seen later in the film under his arm.
When the narrator gives Marla his contact information after they decide to split up the support groups, she asks him his name and a bus passes between them, then the scene cuts away before the bus moves out of the way. For the rest of the film, Marla thinks the narrator's name is Tyler Durden(even though she doesn't start addressing him until near the end) because while the bus was passing between them, he blacked out and became Tyler, introducing himself as Tyler Durden, and then woke up as his main personality sometime later.
When Marla calls the narrator at his new house, he asks her how she got the phone number, and she tells him that he left a forwarding number when they exchanged information.
Several times throughout the film, the narrator says "I know this because Tyler knows this."
When the narrator and Marla first talk at one of the support groups, he tells her that he has been coming to the groups for a little over a year, during which time he has been free of insomnia. Later, Tyler says that he has been living in the Paper Street house for about a year.
Tyler appears in the film in the form of subliminal flashes four times before he is introduced proper in the story:
In the narrator's office at work, when he is talking about how everything seems so far away when one has prolonged insomnia
Next to the narrator's doctor while the doctor is flippantly suggesting the narrator check out the testicular cancer support groups
At that group's first meeting the narrator attends
As the narrator watches Marla walk down the street after one of the meetings but doesn't follow her
Those first three times Tyler appears are before the narrator has cured his insomnia, and at this point the narrator is still subconsciously creating Tyler, hence the brief subliminal hallucinations, while during the fourth time he appears, it is during a brief period when his insomnia returns, and Tyler starts re-appearing during the narrator's waking hours since he is unable to be let out by taking over the narrator's body when the narrator sleeps.
Although not subliminal, the Narrator also encounters Tyler two further times before he is properly introduced in the story: once when he walks past him in an airport, and later when watching television in a hotel room and sees an advertisement for the Pressman Hotel in which a crowd of waiters exclaim "Welcome!" to the camera (Tyler is the waiter on the far right of the television screen).
During his argument with the narrator in the car, Tyler reveals that it was he who blew up the narrator's apartment, despite the fact that Tyler would have had to have done it before they met on the plane; that's because Tyler is a split personality of the narrator's.
Not necessarily. During the airport scene we (through the narrator's eyes) see Tyler driving off in someone else's car. Looking back, the narrator probably just thought Tyler blew up his apartment while he was still stuck waiting on his suitcase.
After the narrator peeks in on Tyler and Marla having sex, Tyler opens the door and briefly talks to the narrator. As the narrator leaves, Marla asks Tyler who he was talking to, because as far as she knows, they're the only two people in the house.
The night Tyler and Marla have sex the first time, which the narrator is unaware of, he has a dream in which he or Tyler (it's hard to tell what with the way the scene is filmed) has sex with her.
Tyler's Space Monkeys threaten to castrate their victims because the Narrator knows exactly how damaging this is - he met all those guys at the testicular cancer group.
One of the easier ones to catch: during Tyler's first fight with the narrator, the narrator hits Tyler in the ear. The next day, the narrator notes that his ears are ringing. During the fight, Tyler never hits the narrator's ears...
This will also relate to the fact that the mind will stop a punch to the face from the same body; the Narrator didn't want to punch himself in the face.
When the Narrator is attempting to explain himself to Marla after the reveal, she's furious with him because, as she put it, "Your whacked-out bald freaks hit me with a fucking broom, they almost broke my arm! They were burning their fingertips off with lye, the stench was unbelievable." The lye part is easily understandable, but the broom part always kind of went over my head until one day it just dawned on me: they hit her because she was standing on the porch, just like Tyler did to them when they were trying to gain entry into the house!
In the book, this is directly stated that she "didn't stay for 3 days" implying that she had to go through the same patience as the rest of them. Good catch.
When Tyler gives "Jack" the chemical burn, he goes into a rant about the nature of God and the image we've created for Him. Then he offers "Jack" two options for the chemical burn: dousing it with water, or neutralizing the base with an acid. On another level, Tyler is asking "Jack" to choose between religion (baptism) or science.
Vinegar (the acid to which you are referring) has biblical meaning as well, though. Jesus was offered vinegar while on the cross, but refused to drink it. In fact, many scholars believe that he refused it because vinegar was used to neutralize pain, and Jesus wanted to suffer without recourse. Interpret this as you will.
One of the recipes Tyler tells the Narrator is dynamite. What did the insurance operator tell the Narrator what his condo was destroyed with a home-made sample of?
In an earlier scene, Tyler and the Narrator are in the bathroom, discussing who they would want to pick a fight with if they could. Tyler says he'd fight his father, while the Narrartor says he barely remembers his father, who left him when he was six. When you realize that Tyler and the Narrator are the same person, Tyler's answer starts to make more sense; the Narrator's dad left him early enough for him not to remember much about the guy, but late enough that his id (Tyler) still wants to beat him up for leaving.
The way "Jack" talks about his father implies he still has some small amount of contact with the guy, enough for the man to offer him worthless life advice and for "Jack" to know about his habit of taking off every six years. Having that kind of relationship with a parent makes you want to beat the shit out of them.
There's the part when "Jack" is getting patched up and Tyler provides his responses for him ("You fell down some stairs." "I fell down some stairs.") The fact that Tyler is a delusion explains this scene perfectly. After all, it would be extremely suspicious otherwise.
Remember that one guy who actually fought back during the "pick a fight and lose" montage? The priest outside the car wash? I just noticed that he is in the background in the fight and project mayhem scenes. Even priests have joined fight club.
Tyler's line that "Self-improvement is masturbation" has meaning on its own, that false improvement (improving because society tell you to) is fake and therefore useless. However, it rings hollow on another level because technically, even his suggestion of self-destruction (basically tearing down and rebuilding) is still technically "improvement," being a change for the better, he just suggests improvement by moving in the opposite direction of the status quo. However, if you think about it, it's more of his brainwashing. If the Narrator did self-improve, it might also have a positive effect on his mental health, causing Tyler to disappear.
And one so small that you may not have even noticed it. That pay phone that "Jack" calls Tyler on? It has this tiny little notice on the front of it: "No incoming calls allowed."
If you care to notice near the end of the movie, you notice that Tyler becomes different. He's wearing sunglasses, a fur coat of sorts and dress shoes with chains on them. Also, when fighting "Jack" his fighting style resembles some sort of Martial arts fighting style as opposed to a rougher street fighting style. Tyler became some exalted and glamorous leader of project mayhem. He became what he taught against.