The cast signed a confidentiality agreement that doesn't allow them to speak and/or explain to the press the meaning of spiders in the movie.
The quote which precedes the film's opening - 'chaos is order yet undeciphered' - is taken from the source novel 'The Double' by José Saramago.
According to the director, Denis Villeneuve, Javier Bardem was offered the lead role but the actor felt he didn't suit the character. Christian Bale was also offered the role and he wanted to do it but couldn't due to scheduling conflicts with other projects.
The song played inside the video store is called "The Cheater". The lyrics are telling a story of a guy who can't control himself with his temptation of sleeping around.
For the scenes in which Adam and Anthony are in the same room, Denis Villeneuve used a tennis ball on a stick as a point of reference so Jake Gyllenhaal would know where to look.