At the film's end, Neil says he's known the Protagonist for years — although it's been just a few weeks from the Protagonist's perspective up to that point — and it's further explained that the Protagonist is the one who established Tenet in the first place. This means at some point in the future, he took The Slow Path and spent years in isolation to travel back far enough in time to get a well-funded and heavily equipped intelligence network and army in place for when he himself gets recruited into the organization he founded.
Inverted cars probably shouldn't work: they need outside air in the same way a person does, or the engine would have to work somehow with fire burning in reverse from normal gas and air in an inverted engine. Setting the gas from the car on fire might also not work for the same reason, it is inverted gas burning with normal air. (Some other parts of the car chase may be iffy, but this troper's knowledge of the cars in question is thin due to not driving much.) Battery powered cars would work, however, since the battery is entirely contained in the car. If only it took place a decade later...
When The Protagonist is holding Sanjay at gunpoint and explaining how he deduced it was Sanjay, Sanjay responds with "a fair assumption..." He wasn't wrong; The Protagonist assumed that the arms dealer was Sanjay whereas in reality it was Priya.
During the restaurant scene, Kat claimed that when she saw another woman diving off the ship, she "never felt such envy" for her freedom. Not for long... that woman was in fact her future self, so in the end, she has her cake and eats it too.
Neil appears way too well-informed as well as much too friendly at his first meeting with the Protagonist, clearly putting him off. The reason is, of course, that from Neil's point of view, he has known the Protagonist for years, and just for a moment Neil has forgotten to wrap his head around the fact that for the Protagonist it's their first meeting. Which promptly turns their friendship into another layer of Fridge Brilliance when it works in the same way as the temporal pincer - Neil is friendly at first but then becomes more distant (as he needs to reign himself in from revealing too much), while the Protagonist warms up to Neil over the course of the film. This is reflected in their arm patches: The Protagonist's red and Neil's blue.
The Sator square is a sequence of five Latin words arranged to read the same up/down/backwards/forwards (SATOR, AREPO, TENET, OPERA, ROTAS). All these words appear in the film:
The antagonist is named Sator ("SATOR" means "seeder" or "founder"; he's the one who kicks off the plot)
The forger is named Arepo (no meaning has been given to this word; but if we were to consider "Opera's" meaning, in the context of the film Arepo would be a "creator/laborer of false works")
Tenet is at the core of the film ("tenet" means a principle that one believes in)
The first action sequence is at a Kiev opera house ("opera" means "work" or "labor")
The Protagonist hides in a Rotas security company container ("rotas" refers to rotation or turning things around)
The two teams' colour coding in the final battle references the phenomena of redshift (for an object that is "going," i.e. moving away from observer from the point of view of the standard flow of time) and blueshift (for an object that is "coming," i.e. moving towards the observer from the point of view of the standard flow of time).
"Tenet" is both a Trust Password and the name of the organization which uses said password. Tenet knows (and uses) time inversion. The word "Tenet" is a palindrome; together with the interlaced-fingers-gesture that accompanies it, the word with its central unrepeated letter also represents the temporal pincer tactics employed by the organisation. Also Tenet sounds similar both forwards or reversed. Which would make it easy to hear if Tenet members meet each other while moving in different directions of time.
The Tenet organization's interlacing of two hands is noteworthy when you look at the final shot of Kat taking her son's hand. Kat represents the present, whilst her son represents the future. Kat's survival through Priya's death is probably the final thing that connects the present and the future.
The final battle might appear confusing with the uniforms making it very hard to tell individuals apart, but this is intentional, as uniforms as well as enclosed vehicles have been used for this purpose several times before: During the freeport raid, the uniforms making it hard to tell people apart is first used to conceal that two men are the same man, once inverted and once in normal time flow. The Fridge Brilliance even becomes doubled when it is eventually revealed who the uniformed man at the freeport actually is.
What's happened's happened. Neil describes this as the mechanics of the world, which can be defined as fate or reality. Fate is what will happen and generally applies to the future, while reality is what has happened and applies to the past/present. Within the context of time inversion, the two views are depicted and interlocked through the actions and dynamics of Neil and the Protagonist.
The Protagonist's fight at Oslo is a self generating time loop. Neither of them started the fight: forward moving protagonist sees inverted man jump out of the turnstile and start wrestling and messing with a gun, inverted man sees protagonist grab him and threaten him with a gun.
When the Protagonist starts breathing in and out at Oslo with Neil, Neil tells the confused official one word to explain it: "Yoga". Obviously, that's not true but it does rhyme with another word that is: Goya. Not a palindrome like many other things in the film, but it does say what the Protagonist is really here for at that point in time.
The first time he was in the Oslo freeport, the Protagonist failed to find and destroy the forged Goya, meaning that Kat was still trapped. The second time, he and Neil succeeded in getting an inverted Kat in and out of the freeport Turnstile, effectively saving her from Sator's control (since Sator left her for dead and thus had no reason to think she would survive).
The main events of the film are visited in chronological order, then reverse-chronological order. The 14th of the month is the day that the Kiev opera siege, the Final Battle in Stalsk-12, and the yacht incident in Vietnam took place. Next is the Oslo freeport infiltration, and the Tallinn truck heist after that. The film visits these events as follows: the 14th, Oslo, Tallinn, Tallinn, Oslo, the 14th. The bulk of the plot of Tenet is palindromic.
After the interrogation, the protagonist says "he would have shot her anyway". Makes sense on its own for a spy movie, but he can also see this from the inverted bullet hole.
Neil brings up the grandfather paradox to explain the mysterious future antagonists' goals. There's another popular idea about time travel known as the bootstrap paradox, in which items or information is present in a Stable Time Loop with no point of origin outside of it. Tenet, Sator and the future antagonists all operate on knowledge gained from observing inverted time states, making the entire film a bootstrap paradox in action.
In Kat's final scene with Andrei on the boat, she makes a point of putting sunscreen on his back, enough to leave a huge smear. After she's shot him, she's able to drag him across the floor much more easily because of the slippery sunscreen; you can even hear it squishing.
The movie offhandedly mentions that Never the Selves Shall Meet is in effect because touching your inverted self would result in something akin to matter-antimatter annihilation. One kilogram of antimatter annihilating one kilogram of normal matter releases the equivalent of 43 megatons of energy. For comparison, that's three times as much as the strongest-ever US nuclear bomb (Castle Bravo, ~15 MT), almost as much as the most powerful man-made detonation ever (Tsar Bomba, ~50 MT), and 3,300 times the Hiroshima bomb (13 kT). The first two examples vaporized entire islands with their detonations, and what the third one did shouldn't require explanation. You wouldn't even need to scale this up to an average human of ~75kg to get a "device" capable of destruction beyond comprehension, and all it'd take to set this off anywhere in the world would be one inverted suicide bomber shaking hands with their normal self at an agreed-upon time and place. Considering how easy it appears to be to invert people, the consequences of this technology falling into the wrong hands become even more nightmarish than they already are.
Which creates some Fridge logic: touching yourself and annihilating creates a grandfather paradox, you would have to touch your inverted self, invert, than touch your original self, but you annihilated after touching your original self and can't do anything after.
A small one, but when Andrei takes off his cufflinks in the bedroom on the yacht, he doesn't put them into the lockbox where Kat's just hidden the Protagonist's gun, as we're made to believe—no, he sticks them into his belt and wraps it around his hand, clearly intending to beat Kat with it.
Kat calls the Protagonist out on his bullshit about Arepo by telling him there's no way he and Arepo could have met, as he's crippled, presumably since Sator had him crippled. Protagonist tells her that they spoke on the phone, to which she responds "he can't do that either". Later in the film, Sator describes his rather complex plan to slice Protagonist's throat and stuff his own testicles inside it. Well, now we know why he can't talk on the phone either—and not just because he's very likely dead.