Do you think there is a unified Imperial culture? Are there traits that neighboring peoples think of as "Imperial," whether they are accurate or not, in the same way we think there are characteristics common to Americans or Japanese or what have you? If so, what are they? Given the great distances the Imperium covers, the slow rate of information communication, and the limited nature of 3I government, how did this unifying culture form, and what sustains it?
Along the same lines, is there an Imperial pop culture? In the 3I, do you see the same media consumed by a large enough percentage of the public to form a common frame of reference such that someone from the Marches and someone from the Solomani Rim could name the same celebrities, quote the same catch phrases, and know the same songs?
No - mostly, though Mike's comments are worth thinking about.
Why no? Two reasons - time and distance. Distance, in this case the week-plus travel time between worlds one to four-light years apart, tends to isolate cultures. Jewell is a couple or three months from Aramis. Aramis is, what, a couple of years from the core worlds? And the Third Imperium is over a millenia old, and the Marches almost as old. Given a degree of isolation and time, cultures tend to evolve in their own unique ways.
Why only mostly? Because there will be similarities based on the fact that they evolved from similar parent cultures, and because interstellar trade and communication will tend to disseminate cultural memes, offsetting the isolation effects to some extent.
Think of 1860's America, the nation an offspring of British culture, but the north and south as culturally different from each other as night and day - and yet similar in language and in certain basic assumptions about government. That took only a couple of centuries and differences in economies and environment.
Think of the Roman Empire, its fall and the evolution of its former parts: Italy, Spain, France, England, Turkey, the Arab world, parts of Germany, and so forth. Some of them kept the bulk of their former culture, incorporating small portions of Roman ways. Some were profoundly affected. All evolved in their own ways afterward - but all share some similarities evolving from their common contact with the culture of Rome.
The Imperium has a homogenizing force in interstellar trade and communication. However, travel times are long and the Imperium has been around for over a millenia - and is composed in places of even older interstellar communities that experienced long periods of being out of touch with each other but that share a common history before that. All of that suggests an Imperium that shares certain broad cultural traditions - that bit about being governed by an Emperor and such, for example, and an apparent common language, or at least a common trade and government language - but that is likely to have profound differences between regions.