To various degrees, all original Xbox Splinter Cell games are compatible with the Xbox 360 console.
In regards to playing original Xbox games on an Xbox 360 system, Wikipedia has a good page on the list of Xbox games compatible with Xbox 360. There is also further compatibility, supported through the Xbox store.
Playing an Xbox Disc in an Xbox 360
In regards to directly supporting an original Xbox game disc, there is an archived list from Xbox.support that lists 510 different titles. According to the Wiki page, this accounts for around 55% of the titles released for the original Xbox1.
- Compatibility is the most important part, and you can check against an archived list from the Xbox support page. If your game is not compatible, there is not much you can do. If you really wish to play your game, see alternate solutions to play the game.
- A hard drive is required to store the relevant emulation data, as well as player data, cache data or any other data used by the particular game you are trying to play.
- Access to the emulation software is required, as each game requires its own emulation software to be downloaded. According to reports, you do not necessarily need an active Live connection, on your Xbox. See below, for downloading the emulation software.
- An Xbox 360 controller will ideally already be available, but it is worth mentioning that you will need to use an Xbox 360 controller to play the game, and can not use an original Xbox controller. See using an Xbox 360 controller for changes you may need to be aware of, due to button changes on the controller.
Downloading the emulation software
You will need to download and install emulation software specific to the game you are playing on to the Xbox 360 console. To quote the Wikipedia page,
"These profiles can be downloaded via the Xbox Live service or from
Xbox.com and burned on CD or DVD recordable media. They also were
routinely distributed as part of the content on the Official Xbox
Magazine monthly discs (which were discontinued in 2012). The profiles
are downloaded as a single bundle, thus only needing to be applied
once per update (if doing so offline), or are automatically downloaded
the first time a compatible game is inserted."
While Microsoft have been known to remove content from their stores, as they deem it "outdated", this suggests that there is an 'offline' option for acquiring the emulation software. If you wish to play a particular game that was supported, previously, you might be able to find an external source for the emulation software, and burn it to a disc.
Splinter Cell support
All three of the original Splinter Cell games are listed as supported titles, across most regions; the original Splinter Cell does not support NTSC, and only Pandora Tomorrow supports wide screen. Apart from that, the page lists the following known issues:
- Splinter Cell: "DLC is supported. Keypad colors do not appear properly when using the thermal vision goggles. There is no sound in certain levels. There is a notable drop in frame rate in some areas."
- Splinter Cell - Pandora Tomorrow: < no reported issue >
- Splinter Cell - Chaos Theory: Minor graphical corruption on the save/load screen. Noticeable frame rate drops occur throughout the game.
Splinter Cell: Chaos Theory HD support
As you mention, Chaos Theory is actually available from the Xbox store, to download to an Xbox 360 console. When I look on the Brazil Xbox Store, using the same search parameter I used to find it on the Australia Xbox Store, I had no luck.
This, to me, implies that the game is not accessible from your current location. That said, I base that off not having access to particular content from Australia, which I can fix by logging into an account with its country set to a location that did have the game listed.
I only used an alternate account to access the independent marketplace, for a game that was actually made by a local developer. I have been told that bypassing the access rights in such a way is strictly monitored, and can lead to account closure, but this never happened to me. This might very well be due to me only accessing content that was simply not made accessible to my location, where you might have concern regarding the more critical issue of actual rights to distribute.
Using an Xbox 360 controller
It is worth addressing the controllers, as there is a slight change in button mapping. The original Xbox controller had two additional buttons, 'white' and 'black', grouped with the A, B, X, Y buttons. Additionally, the Xbox 360 controller has bumper buttons, LB and RB, which the original Xbox did not.
If you had not already guessed, the new bumper buttons take the place of the old black and white buttons, as far as input is concerned. Any reference to 'black button' on an original Xbox game refers to the right bumper button, RB, while any reference to 'white button' on an original Xbox game refers to the left bumper button, LB.
Alternate Solutions to Play the Game
If the particular title you wish to play does not appear as a compatible game, or for any reason you can play it through the official emulation, there are alternate solutions you may wish to look at in order to play the game.
- Acquire an original Xbox console: Depending on your location, you might be able to easily acquire an old second-hand Xbox console. If you want something cheap, try a second hand store before you try a game traders. If you want more reliability, a game traders would probably be your bet. If all else fails, try eBay or Gumtree.
- Use computer-based emulation software: A controversial topic, no doubt, but in certain countries2 you are legally allowed to download emulation software to your computer, for the purpose of running video games that you legally own. Three more-known emulators are Xeon, Cxbx and Dxbx.
- Remastered versions: It is worth making sure that the game was not re-released, as you may have a more accessible and potentially higher quality version of the game available to you. For example, if you have access to a PlayStation 3 console, all three Splinter Cell games were remastered and released to the PlayStation store. You can purchase Splinter Cell, Pandora Tomorrow and Chaos Theory separately, and download them to any PlayStation 3 console using the account you used to purchase them.
1 The Wikipedia page last reported "461 (titles) .. Approximately 51% of the original Xbox library", and since the update, several more games were added, so I worked out the new percentage based off the original percentage listed with Wikipedia.
2I am not a lawyer, but I know that in my country (Australia), I am allowed to legally use emulation software to run games I own through my computer. I am also aware that in particular countries, the exact same practice is illegal, but then again, some countries outlaw playing video games all together. If you are not sure, find out, or ask on Law.