Legendary Pictures, Syncopy Inc., and Warner Bros.
The deets: Inception was produced by Legendary Pictures, produced by Syncopy Inc. and distributed by Warner Bros.
Syncopy is Christopher Nolan and (his wife) Emma Thomas' production company: they created it in 2001. The first film released under the Syncopy label was Batman Begins in 2005 and Syncopy has continued to produce all of the couple's ventures and a (very) few films directed by others.
Since its first release, Syncopy has partnered with Legendary Pictures—a relatively new production company founded in 2000—which has been in partnership with Warner Bros. since the release of its first Syncopy movie in 2005. Fun fact: both Syncopy and Legendary have similar, complicated maze-like logos. Syncopy features its company name embedded in a standard maze while Legendary uses an ancient Celtic shield knot which apparently represents eternity, fidelity, and unity.
As for the films themselves, Legendary states that it has "a targeted focus on the powerful fandom demographic." They are not kidding. With the release of film adaptations like Syncopy's Batman trilogy and other super hero/comic book releases like Man of Steel and Sucker Punch, and even classic reboots like Jurassic World, Legendary is big on making movies for specific fan bases.
This strategy of "in-crowd hype" has also worked in terms of marketing where lots of mystery is employed, especially when Syncopy is at play. Viral marketing like that was employed for Interstellar, Jurassic World, and Inception itself.
To promote Inception there were posters about mind crime, a short animated clip of a spinning top, a dream machine replica, a video game, and other random websites, all of which were more intriguing than revealing. Sometimes it's what you don't know that draws you to a movie…but the fact that it was directed by Christopher Nolan probably helps too.