I know they can with Android devices. Many unlocked Android devices that are released before a carrier officially launches them come without bloatware that is somehow an integral part of the OS by the time it gets to you.
I know that the Samsung Galaxy S II unlocked version has a couple special app pre-installed by Samsung, but I guarantee that when AT&T picks it up, it will come with AT&T Code Scanner, and AT&T Navigator etc... Without rooting the phone, you cannot uninstall applications.
So, yes, it does seem that carriers are able to specify changes they want made to a phone's OS before it is released.
In terms of apps, sure - this is true of any phone. However the menu structure / core functionality of the phone? I'm just trying to work out whether this is an app in effect, or a deeper level of customisation.
They Can on BlackBerry as well. For example, AT&T has removed the Call Barring feature from the software builds on they're devices in order to force customers to use AT&T Services.