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The Assassin Order

"Nothing is true, everything is permitted."
Also known as the Assassin Brotherhood, they are an ancient group dedicated to preserving the freedom of mankind from any who would threaten it. Their main enemies are the Templar Order, who seek to use the Pieces of Eden to dominate the world. They have long operated in secret, only occasionally recruiting exceptional individuals into their ranks. By 2012, the Assassin Order is severely depleted and on the run, desperately trying to stop the Templars from achieving their final goals and a massive potential global catastrophe from a massive sun flare.
Animal Motif: Eagles most prominently.
Anti-Hero: Type III at best and Type V at worst, but they're usually in the middle as a solid Type IV. They’re merciless and ruthless and will not hesitate to kill anyone affiliated with the Templars or people who they feel stand in the way of the greater good. But by the same token they will Never Hurt an Innocent (well, usually anyway) and they also have genuinely good intentions
The Anti-Nihilist: The Assassins; "Nothing is true, everything is permitted."
Altaïr Ibn-La'Ahad: ...laws arise not from divinity, but reason. I understand now that our creed does not commend us to be free - it commends us to be wise.
Ezio Auditore da Firenze: ...merely an observation of the nature of reality: To say that nothing is true is to realize that the foundations of society are fragile, and that we must be the shepherds of our own civilization. To say that everything is permitted is to understand that we are the architects of our actions, and that we must live with their consequences, whether glorious or tragic.
Black and Gray Morality: Usually the gray to the Templar's black.
Blade Below the Shoulder: Their trademark Hidden Blade, which they have used into the modern era (although its more ceremonial by that point).
The Combat Pragmatist: Frequently.
The Determinator: They have a bit of a habit of including them, and the order as a whole probably fits this too.
The Fettered/The Unfettered: The Assassins strive for Freedom/Chaos, yet live by a strict moral code that defines all their actions. The Templars, in contrast, seek Order/Law, yet have absolutely no moral restrictions on their behavior and are free to use any and all means (up to and including mass murder) in pursuit of their goal: this leads to quite a few ironies. Ultimately, both the Assassins and Templars believe that "Nothing is true, everything is permitted". But to the Assassins the phrase is descriptive whereas to the Templars it's proscriptive.
Good Is Not Nice
The Hashshashin: Their front in the Third Crusade.
Hidden in Plain Sight: One of the common Assassin methods of getting through missions unnoticed, through whatever means necessary. It is, in fact, part of their Creed from the Middle Ages.
In the Hood: A recurring part of their uniforms over the ages.
Leap of Faith: The trademark skill of the Assassins, along with the Hidden Blade. All recruits are required to perform this as part of their initiation as full Assassins.
Man in White: Their colors are white and red. The original reason for the former was so the assassins could disguise themselves as average priests, who also wore white hooded robes.
Light is Good: But not pure.
The Mentor: A title given to the overall leader of a Assassin Guild, and eventually all Assassins worldwide.
Never Hurt an Innocent: One of their tenets as part of their eponymous Creed. However, they're willing to make exceptions for Unwitting Pawns of the Templars and, in some cases, witnesses of their crimes.
Order Versus Chaos: Technically the chaos to the Templar’s order, but here the “chaotic” group are (usually anyway) the good guys.
The Purge: Beginning in 2000, the Assassin Order has been in hiding and on the run after Templar sleeper agent Daniel Cross killed the current Mentor and gave the Templars the location of dozens of Assassin bases.
Sliding Scale of Idealism Versus Cynicism: Believe it or not, they're the idealists.
Well-Intentioned Extremist: The Assassins are willing to ally with evil people and groups if forced to, and will kill those being manipulated by the Templars without knowing it, in order to safeguard humanity.
Taken further in III. Desmond asks his father, William whether the Assassins have ever tried to do something to a Templar agent like the Templars did to Daniel Cross or Lucy Stillman. William admits they did, but that they just weren't as successful due to their agents either being too weak to resist being turned or too strong to appear to be turned.

The Harlequin (Cahin)

One of Cesare Borgia's assassins, Cahin was a corrupt French nobleman that dressed as a harlequin. Together with his sister Caha, he would lure his victims to their deaths while putting on a show for them.
Aristocrats Are Evil: And also totally mad.
Died in Your Arms Tonight: Slips away while his sister cradles him, seconds before she too, is killed.
Downloadable Content: Only available as a persona through pre-ordering certain editions, or after a system update.
French Jerk
Impaled with Extreme Prejudice: Although he is fast enough to keep up with them for at least a little bit, Cahin can't hope to match multiple assassins with two daggers and gets stabbed multiple times in the gut before going down.
Knife Nut: Wields two rondel daggers.
Malevolent Masked Man
Monster Clown: This guy could give The Joker a run for his money.
Nice Hat: A typical jester's hat. Looks vaguely like a croissant if you think about it...
Running Gag / Big Lipped Alligator Moment: There's an achievement called "Clowning Around" that involves fighting a jester using The Harlequin's character model at six different locations. He'll be doing some acrobatics on the street, and once Ezio gets close enough, he'll engage him on a fist-fight out of nowhere. Whether the Harlequin in question is Cahin or not is debatable, but seeing as you must defeat him with your bare hands (killing him does not count towards the achievement, and that includes dropping him on water), it's possible that Ezio fights Cahin at six different places instead of fighting six different Harlequins.
Professional Killer
Sibling Team: With his sister, The Hellequin.
Theme Naming: With his sister. The phrase "cahin caha" means "willy-nilly" in French.

The Templar Order

"May the Father of Understanding guide us."
An ancient order founded by Cain, with his mark being used as their symbol. They have warred with the Assassin Order throughout history over control of the Pieces of Eden, in the hope of establishing a New World Order that would eliminate The Evils of Free Will, while also putting them in charge of the planet. They have operated under many names, including SPQR, the Château Clique, the Secret Chiefs, and Abstergo Industries in the 20th and 21st centuries. During their eternal war with the Assassins, they have been both forced into hiding or dominant, with the Templars currently holding the upper hand in 2012, with the Assassin Order scattered and being systematically hunted down.
Ancient Conspiracy: Up to Eleven
Anti-Villain: Many of them are Type II Is.
Arch-Enemy: To the Assassins Order
Assimilation Plot: Their ultimate goal.
Black and Gray Morality: Usually the black to the assassins gray.
Covert Group With Mundane Front: Several, with Abstergo being the biggest one.
Even Evil Has Standards: Modern day Templars are disgusted by the state of the Order during the Renaissance, since most of the members then were only out for personal power and had abandoned the original Templar ideals.
The Evils of Free Will: Their main contention point against the Assassins' Anti-Nihilist.
Fascist, but Inefficient: Whenever the Templars took full control of an area in historical times, they tended to run it into the ground due to their "keep the people weak" policy weakening the nation as a whole, making it extremely difficult to achieve their Utopia.
The Fettered/The Unfettered: The Templars, in contrast to the Assassins, seek Order/Law rather than Freedom/Chaos, yet have absolutely no moral restrictions on their behavior (as opposed to the eponymous Assassin's Creed) and are free to use any and all means (up to and including mass murder) in pursuit of their goal: this leads to quite a few ironies. Ultimately, both the Assassins and Templars believe that "Nothing is true, everything is permitted". But to the Assassins the phrase is descriptive whereas to the Templars it's proscriptive.
Humans Are Morons and the real monsters: Many of their members tend to think this way, in contrast to the assassins' general belief that Humans Are Flawed but ultimately worth saving.
Misanthrope Supreme
The Knights Templar: Their front in the Third Crusade, hence their name.
Knight Templar: They also fit this trope in personality too.
Mega Corp: As Abstergo Industries.
New World Order: The goal.
Order Versus Chaos: They see themselves as the Order to the Assassins Chaos. Too bad their “order” always equals something along the lines Fascist, but Inefficient.
Sliding Scale of Idealism Versus Cynicism: The cynics to the Assassins idealists
Utopia Justifies the Means: What the group hopes to achieve.
Well-Intentioned Extremist: What most of them like to think and portray themselves as, although many are just in it for the power.