All-round King

This variant combines several variants with different goals about the kings: an Anti-King chess, where player must keep his king under opponent's attack; Escaping King chess, where player must move his king out of opponent's attack; Surrendering King, where one must move king under opponent's attack; and the normal chess, where you have to protect your king.
I'm not sure about common names of the second and the third variants, but they certainly was ever invented by someone. So, i'll call them "Escaping King" and "Surrendering King".

Rules

Game starts with the Escaping Kings. Each one is located on positions of opposing king in normal game: white is on E8, black is on E1, being attacked by queens.
Escaping Kings can't capture opponent's pieces, only friendly ones.
There is a forced castling: if opponent's king hasn't moved yet, player may castle it with own rook. It's only time when you can move opponent's Escaping King. Since this king is always under check, there is no usual castling restrictions.
To win this phase, king's owner must move it out of check, so that opponent can't attack it again on next turn.

Next phase is the Surrendering King. Position is not changed, but the king, wich have just escaped, turns into the next formation (the other king remains as it was).
To win this phase, player must put his king under opponent's attack, so that opponent can't remove this attack.
This king can capture opponent's pieces, not friendly ones.

Third phase is normal king.
Here king changes owner. Therefore, one player can have two kings (in different formations), while other player don't have kings at all.
Now player must simply checkmate this king.
Normal king captures opponent's pieces.
Note that it's possible to skip this phase to the fourth: it happens if in the second phase you'll put king under unescapable attack of both opposing and friendly pieces.

Fourth phase is the Anti-King.
King's owner is same as in normal king phase. Player must remove attack from this king, so that opponent can't put it back under attack. Once this is done, player wins the game.
Anti-King captures friendly pieces.

If there is a danger about both kings (e.g., normal King under attack, while oppnent's Escaping King has moved out of attack), player must solve both problems if possible. If player must make choise, wich one to solve, player must solve one, leaving the second king in illegal position. This king goes to next phase.

If you want, you may change the order of the phases, but i think this order is the most optimal. It alters the phases, where player gets advantage by capturing opponent's pieces, and the phases, where capturing is disadvantageous, and it provides the challenge: player have to choose between gaining advantage now, or keep opponent's pieces for later advantage.

There are 4 additional phases: same as the original four, but with friendly-fire. Friendly pieces checks the king, not opponent's. Game can be played with 8 phases.
Sides of pieces, king can capture, swaps. E.g., friendly-fire escaping king can't capture friendly pieces, but can capture opponent's.
When you choose the order of phases, remeber that some phases must start with king under attack of certain side, and some must start not under attack of certain side.
And, if you want to play with friendly-fire "normal" King, remember that it's almost impossible to force checkmate him with otherwise usual rules and pieces. So you have to use something special to force opponent to attack his own king - e.g., Rococo Swapper.

Probably, it's better to be played with more pieces on board with not very big size, especially with more phases.
Hoppers, like xiang-qi cannon, will give special sense.