If you really wanted to do it you could. I am surprised to hear that there is no adaptoid that will convert a dreamcast controller signal to xbox. Anyways, if that method is a no-go then you can hard mod the controller yourself. The basic idea is to open a Xbox controller and a dreamcast controller and connect each button and analog stick to the corresponding contact point. It is probably more work than its worth, but apart from the payoff of having a cramp-free controller to use, you'll also have the extra "OMG 1337 HAx0rz!" appeal.
And chicks dig that.
Well...not attractive chicks but chicks nonetheless!!
So bust open both controllers (figure of speech, of course

) and find your wiring points. I have some pics that may help you here-



Take what you can from these.
A quick google search yielded this website which shows how dreamcast controllers send out their signals. A Madcatz controller is seen, but with some ingenuity I'm sure you can learn from this website and take the next step. Check these out to get a feel for what kind of task this is.
http://www.darkravenwind.com/cdvision/images/newpad.jpghttp://www.darkravenwind.com/cdvision/madcatz.htmIn a nutshell, you would have to wire the buttons for A B X and Y to the Xbox controller's points for A B X and Y, Start, as well as the triggers and analog stick(s) and D-Pad. Kind of tedious work, but possible.
That should at least point you in the right direction
happy modding