Can I use Xbox 360 controllers with Xbox Classic (first generation of Xbox)?

Searching around, I don't see any kind of adapter for an Xbox 360 controller to an Xbox port. I don't think the original Xbox supported any wireless controller either. The Original Xbox certainly had some sort of proprietary port for it's controllers, but I've read that it's just a standard USB in a weird form factor. Third party modifications may be possible, but would be quite a hassle to try.

According to the Wikipedia page for the Original Xbox controller:

Neither the original Xbox controller nor Controller S are compatible with the Xbox 360, and the Xbox 360 controller is not compatible with an original Xbox.

I think one of the biggest problems between the platforms is the software used to interpret the inputs of the controllers. The Xbox 360 controller uses the XInput API, while the original Xbox's controller likely used something else (it's not mentioned on the webpage for XInput anywhere), meaning even if you were able to somehow get an Xbox 360 controller to connect to an Original Xbox, it probably wouldn't be able to read any input.