You can go that way. I would probably approach it as a control hack.
Before going too much further you do have the hardware approach, which has the added bonus of only working for that controller, working in other games and working on unmodded xboxes (I don't know who would have such a thing but I note it anyway).
General idea is desolder the vertical pins of the stick (though x axis here might mean those, might as well do both though as plenty of games still suffered from cretin camera for vertical stuff), swap them around and there you go. Can even chuck a few more wires and a suitable switch to have it go back and forth depending upon your whims. Even mediocre electrical skills can probably get this done an evening project, without skills in software this is on the more advanced side of things, especially for the xbox as debugging that is a pain and emulation is not up to too much (though getting better rapidly).
Anyway software based stuff, sadly I don't think the original xbox has enough of a kernel that you can do it there (you can on the 360, 3ds and obviously PC, anything with a hacked kernel running in the background is probably handling controls for the game so you can attack that for your control modding purposes on those) Control hacks then. Several approaches depending upon what is going on.
1) Somewhere in memory is every frame or so the state of the buttons. Good coders will not use this directly but instead copy once a frame and have everything operate from that. This is as it dodges some issues with bouncing -- switches are mechanical devices rather than clean ideal binary devices and might display everything from invalid voltages to flickering between values as things settle, which is annoying when one part of an action sees the button pressed and acts accordingly but something else checks and finds it different despite only being nanoseconds later and then we get oddities, as more than one button press happens in the average game the chances of controls failing you (a bad thing in game design) you debounce things.
Anyway sticks are analogue so you get analogue values rather than on or off. Same idea though as you are either going to do the reverse of the value the system presented (if it is say between -127 and 128 then if it is minus then make it positive, if it is positive then make it negative, or if it is a simple range then... yeah), or change what the game reads (in horrible psuedocode READ button state, if X axis is between ? and 0 do according movement, if between 0 and # then do according movement though the opposite of the last one, easy enough then to make it look at another stick, or take the other path that normally gets selected by the other direction).
If there is any kind of button mapping in the game, even "premade selection we think you will like" then there might well be further options. If there is a sensitivity option it might also get in the way or be a factor in the distance moved.
I don't know what we have for xbox hardware listings -- homebrew sdks might know, possibly commercial ones (by this time they had gone away from hardware documentation and instead gave devs libraries), emulator source code and wikis. There is a possibility you can find such things with cheats but eh. If you wanted to continue with the camera value and work backwards from there for anything altering it (will need a debugger you can set breakpoints in, find the camera and then set a break on write, move the stick and now we are off.
https://www.romhacking.net/documents/361/ is for the GBA but the same principles apply for basically everything when it comes to tracing as it is known. Hopefully the camera is in part of memory your cheat program can see and manipulate as well -- the xbox was basically a PC so you do have directX after a fashion which is fairly fully featured 3d rather than the more primitive stuff seen in other devices of earlier years (or later handhelds -- the DS was still pretty primitive as these things go).
If you need to change any tutorials or on screen prompts as part of this then that lands us in text and graphics hacking world, possibly audio as well as it was a space year where you could afford to have hours of audio without breaking storage limits.