These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Designated Hero: Kamui at first due to being a Jerk Ass who avoids contact with anybody. It should be noted that Yuzuriha and Sorata actually understand why he acts like that and Hinoto even reveals that he could actually destroy mankind depending on his development despite her visions of Kamui as messiah figure. Kamui finally becomes more likable when opening up to Kotori.
Ho Yay: C'mon, it's CLAMP. The most prominent case Kamui and Fuuma is later turned Foe Yay, natch. There are other common matchups as well, including Subaru and Kamui (mind penetration and Big Brother Mentorship FTW) and Fuuma and Seishirou (mostly for the symmetry, it seems). Seishirou and Subaru are far enough out of subtext and into text that Ho Yay doesn't really apply anymore.
It fits pretty well in shoujo standards, however, considering the storyline's strong focus on emotions and drama. That's very proper of shoujo mangas.
What Do You Mean, It's Not Symbolic?: This is essentially the Gospel According to Ageha. The major alteration from Judeo-Christian eschatology is the dualism, with the two sides evenly matched and only a slim moral advantage on the part of the Dragons of Heaven.
Also, Kamui was Shinji Ikari before Shinji ever really existed. He was HOTLY bashed and loathed by fans for daring to cry and angst over his shitty destiny and the terrible breaking of his, Kotori and Fuuma's bonds.
The Woobie: Kakyou, Kotori, Subaru, Kamui, Yuzuriha when Satsuki decides it's time for a very literal Kick the Dog moment and kills Inuki... a good chunk of the main cast, really. Not so strange when you consider that they're Team Keep Humanity Alive. Among the fandom, Subaru is the undisputed King of Woobieness.
Kakyou Kuzuki is a Dragon of Earth. Team Kill Humanity to Save the Earth isn't entirely without Woobies, you know? Outside the movie, pretty much everyone is a Woobie or just plain Affable. Fuuma's theme is granting wishes (read: Killing people who want to die), the Sakurazukamori assassin is a position apparently passed on by death at the hands of the one the current assassin loves (Seishirou killed his mom), Satsuki was used by first a father and then corporate businessmen who want to exploit her. Kusanagi is perhaps the only one on the Dragons of Earth who takes his side (though halfheartedly) because he's a nature-lover. (Oh, and he's ready to Back Stab FUUMA to save Yuzuriha.) Nataku hardly has a past at all because he's a clone. That leaves Yuuto, who's incredibly cheerful and friendly who's pretty much in it For the Lulz, and even then he wasn't fully evil as he has a sister he misses and genuinely cares for and refuses to take advantage of Satsuki's crush on him when he could've perfectly done so.
Nataku's case is ambiguous, since he might not have a proper past since he's a clone but in the manga what's left of the person he was cloned from's memories play a very important part in his very spoileriffic and sympathetic death. You could even make a case for the anime, as Fuuma absorbs Nataku's body to heal himself after Sorata almost kills him because of his Wish and former life's memories, in a sense.
Subaru was a Woobie before X even started, having lost his twin sister at the hands of their best friend / mutual love interest, Seishiro. Yes, Ho Yay and Foe Yay in a single package.