Americans Hate Tingle: The anime, definitely. It was critically acclaimed in Japan and even won 2 "Best OVA" awards. But viewers everywhere else found it a disjointed mess of a Mindscrew. When the novels were released in English, they received some pretty positive reception, and it's common to find non-Japanese readers saying that the anime made a complete mess of the novels. Japanese fans don't seem to think so; general consensus among them appears to be that the anime is a perfectly acceptable alternate version of the story.
Most clips & AMVs of the show on Youtube are uploaded by Japanese users. Again, this is on Youtube, not Nico Nico Douga.
"Apotheosis", the last track played in the entire series as Rei makes his Heroic Sacrifice to collapse the Passageway. It's quite dark and depressing, but still awesome, and considering the show's limited music, arguably the best track in the entire OST.
Genius Bonus: Probably half the technical terms and acronyms used among the pilots in flight (and even terms that simply appear on computer screens) won't make much sense unless you are a pilot yourself, know someone who is a pilot (preferably a military pilot), or look it up.
Ho Yay: There's a reason why the OVA is nicknamed "Brokeback Air Force."
In an interview Jack's voice actor even comments on poor Jack ending up with his heart broken.
There are a couple of occasions early in the first novel when Jack jokingly tells Rei to "break up with Yukikaze and come work for me." The novels, though make it clear he views Rei as a friend and more importantly, he stops telling Rei things like the above once he accepts that Yukikaze is notJust a Machine.
Narm: In the second novel, Ansel Rombert calls the JAM human copies "Jammies." Say it out loud and try not to think of pajamas.
Strawman Has a Point: Despite his Jerkass & Eagleland tendencies, Andy Lander does bring up some good points regarding the FAF's relations with the UN, such as the UN forbidding the FAF from growing food. This makes them wholly dependent on a UN food exchange program and may well be a means to prevent the FAF from rebelling against Earth. Rei is mildly surprised at this: he knew the FAF imported food but had no idea it wasn't allowed to actually produce its own food.
Uncanny Valley: Rei immediately senses something off about nurse Marnie & Major Yazawa at TAB-14. Yazawa's uniform is the wrong shade of color per FAF regulations, & Marnie's disposition is too cheerful. They are both JAM copies with minor flaws in their assembly.
The Woobie: Jack, oh poor Jack. And arguably, Copy Tom "Tomahawk" John.