Castor and Pollux are the twins from the constellation Gemini, the twins. On the opposite side of the star calendar is Sagittarius, the Archer.
Castor and Pollux's last name is Troy, referencing the Trojan Horse, which is effectively what the face transplant surgery turns Castor and Archer into.
The blood types for the two main characters (i.e. Sean Archer and Castor Troy) are mentioned in the movie, and at a subtle level they reflect their antagonistic nature. Archer's Blood type is O negative, while Troy's is AB positive, these two blood types are as different as you can get. O negative (the universal donor), reflects Archer's role as a policeman dedicated to serving the community, while Troy's AB positive (the universal recipient) suggests someone who takes from society without giving anything back.
Why does nobody catch on fully that "Archer" is suddenly a lot more carefree and happy? It is implied that before Castor killed Michael, Sean was actually a very fun, friendly and happy guy, they might assuming it is because all the weight and baggage from trying to catch his son's killer has been lifted.
Sasha having no idea "Castor" is actually Archer, Why does she ask him to look after their son? because she thinks "Castor" is starting to have more emotional maturity, and his stint in Erehwon might have "scared him straight" somewhat.
