Synecdoche, New York is a complex, multi-layered piece of cinema, and you can see it however you wish. (Or decide that it's all pretentious nonsense, if you want!) Below are some possible interpretations.
It's often unclear which scenes and events are "real" and which are dreams or imagination, an intentional effect caused by the disturbed state of Caden's mind. This parallels and contrasts with the play directed by Caden, Death of a Salesman, in which aging patriarch Willy Loman holds imaginary conversations with his deceased brother and younger versions of his sons - but in that play, it is at least certain which events are "real" and which are figments.
The representation of time is also intentionally confusing. On the first day, a radio announcer says it's September 22. But Caden's newspaper reports that Harold Pinter has been awarded the Nobel Prize in Literature (making it October 14th, 2005), it's Christmas when the sinks smashes Caden's forehead, New Year's on the ride home and March in the ophthalmologist's office. Kaufman afforded his film a dreamlike quality by playing with the representation of time throughout.
Key to understanding the film is the title: a synecdoche is a figure of speech where a part refers to a whole, or a whole to a part, for example, "hired hands" for "labourers", or "the law" for "the police". Theatre is a synecdoche, where the events on stage stand for an entire world - and conversely, the world can be seen as just a theatre, and life as just a play - as Shakespeare said, "all the world's a stage, and all the men and women merely players." The warehouse in which Cotard builds his replica of New York City (a simulacrum, representation, as per the title of the work, Simulacrum) gradually becomes a synecdoche of his life, and it becomes unclear whether the events of Caden's life are being acted out on stage, or if the events of Simulacrum are being acted out in Caden's real life. This reaches a final irony when (as far as we know) the play is never performed before an audience, but Caden's life is - in the form of the film Synecdoche, New York!
An interesting contrast exists between the increasingly tiny works of Adele and the increasingly enormous work of Caden's. Director Charlie Kaufman said, "In [Adele's] studio at the beginning of the movie you can see some small but regular-sized paintings that you could see without a magnifying glass ... By the time [Caden] goes to the gallery to look at her work, which is many years later, you can't see them at all. As a dream image it appeals to me. Her work is in a way much more effective than Caden's work. Caden's goal in his attempt to do his sprawling theater piece is to impress Adele because he feels so lacking next to her in terms of his work. Caden's work is so literal. The only way he can reflect reality in his mind is by imitating it full-size .... It's a dream image but he's not interacting with it successfully."
The actors playing real people begin to fill these people's roles in reality. Sammy Barnathan (the actor who plays Caden) claims to have followed Caden for years (and is visible watching him on at least three occasions earlier in the film) and in a sense he has, as the character is a fictional observer on Caden's whole life.
Fate - and the importance of accepting fate - plays an important role in the film. Cotard's deteriorating physical condition means he has to confront his own death and accept mortality. He reminds his employees that they will all die too, eventually, and must come to terms with it. Much of life is spent in denial as this - note how Adele tells Olive that she doesn't have blood, denial of the body being a way of denying mortality. Cotard is forced to see everyone as a decaying and dying body - note how he imagines his counsellor having a progressively nasty skin disease. Hazel accepts her fate casually - she even purchases a house which is (bizarrely) on fire, and lives in it for years even though she knows she will die of smoke inhalation (and she does). In a sense, life is like buying a burning house - we know it will be destroyed, but accept it anyway. Hazel's relationship with Caden is similar: she knows he is deeply disturbed and the relationship cannot last for long, but pursues it anyway.
A script is a particular kind of fate, in which virtually all one's actions and words are predetermined, and only tiny variations in tone and attitude are permitted. As a director, Caden works with the writings of other people, and can only add superficial changes to their work, for example casting young actors as the elderly lead characters of Death of a Salesman in order to show that mortality is universal, an alteration the purpose of which his parents (and probably most of the audience) missed. But Adele, as a painter, can create new works, which makes Caden jealous. He attempts to create his own script, but ends up just copying real life. Even his relationship with Claire follows a very similar "script" to his relationship with Adele: they have a daughter and gradually grow apart, Caden retaining an emotional attachment to Hazel. Even the ZIP code of the film's location (Schenectady, NY) follows a predictable pattern - it's 12345! Caden ends by following a stage manager's directions in life via earpiece, even obeying her final command: "Die."
Additionally, the cinema audience are of course also watching a scripted performance, which seems to be an unpredictable series of events, but was actually written and edited and set down on film long before it was seen by viewers.
Hazel's books also have symbolic resonance - Swann's Way is the first volume of In Search of Lost Time, an epic novel by Marcel Proust in which the author explores and interprets the events of his own life. A character in it is named Dr Cottard, who is based on Dr Jules Cotard, a French neurologist who described the Cotard delusion, a patient's delusional belief that they are dead, do not exist or do not have bodily organs. The main character of Synecdoche, New York is also named Cotard, and perhaps he believes that humans have the delusional belief that they are alive and that they exist!
Hazel also reads Franz Kafka's The Trial, a novel about a man named Josef K. who is arrested and told he will have to face a trial, but is not told what he is accused of, and explores a vast and incomprehensible legal bureaucracy before finally being sloppily executed for no clear reason. This could be compared to Cotard's view of the world: one is placed into life for no reason and struggles to understand a vast and incomprehensible world, and finally dies meaninglessly.