I would suggest that the X360 is known mostly for:
Gears of War
Call of Duty
Ghost Recon
Madden
Oblivion
Rainbow Six
Fight Night
Perfect Dark
Lost Planet
Dead Rising
Need for Speed
Saints Row
Crackdown
In other words: Mostly shooters, EA staple franchises, sandbox gang-land games, and action in general, with Oblivion being the main exception. This was probably what MS wanted in order to hook their traditional fanbase at the beginning of the lifecycle: People that love the Xbox brand enough to spend $400 on a console. They've been very successful with this.
Now you can slowly see them bringing out stuff that might appeal to the different segments like Forza 2, Guitar Hero 2, Blue Dragon, Banjo 3, and Mass Effect. It's hard to say whether they can broaden the X360 appeal with these titles, but it probably can't hurt.