The particular variant I favor is Archoniclastic Augmented Chess, in which the "stronger than normal" pieces are defined by the rules of Augmented Chess. I like this choice because the game can be played with different armies.
By contrast, if the pieces standing on their own color became stronger by following the rules of Cylindrical Chess, the strengthening would not be value-preserving, and Archoniclastic Cylindrical Chess could not be played with different armies.
It seems self-evident to me that Archoniclastic Augmented Chess, and in general most types of Archoniclastic Chess, must be good games and playable ones, so I haven't bothered with much playtesting. There is one problem, which is described and solved in the next section.
What about Archoniclastic Progressive Chess? It's crazy, but probably playable. If you move a piece that's on the wrong color, you can make only one move that turn; if you move only pieces standing on their own color, you can make one more move than your opponent just made; and when you make just one move in a turn, you reset the limit so your opponent can make at most 2 moves next turn.
Archoniclastic Mega-Chess! Perhaps that's unplayable? No, no more so than the basic game.
Archoniclastic Shogi! Because the squares aren't colored, you can't play it! So there.
In Archoniclastic Augmented Chess, if a colorbound piece is augmented by a non-colorbound augmenter, there is a problem. If you use the non-colorbound power, you land on an unfriendly square, which means that your piece is no longer augmented, in other words it's now colorbound and must stay on unfriendly squares!
You can reserve the color-changing power as a threat and thus get some value from it, but it's not nice; or, you can choose an augmenter which is as colorbound as the original piece.
There aren't enough colorbound augmenters! How do you augment a BD or a FAD?
Therefore, for the sake of Archoniclastic Chess, two new augmenters are added to Augmented Chess. (The player will still have the right to choose BDW or FADW, even though there is a problem with these choices.)
The Langskip is a long Barc: it makes the (1,3) jumping move, wide forwards and narrow rearwards. For example, a White Langskip on e4 could go to b5, h5, d1, or f1. The funny notation is fsLbbL.
The sL is a wide long Knight (or Camel): from e4 it could jump to b5, h5, b3,or h3.
Note that the King Crab cannot be used. A White ffLbsL on e4 could jump to d7, f7, b3, or h3; and augmenting the FAD with it allows the winning move 1. FADffLbsL f1-d3 with mate threats at f5 and d5, plus a material-winning threat at b5 (attacks a8, and depending on what army Black has, this may win big).
For the same reason, the Narrow Long Knight (fbL) cannot be used.