Is it ever legal for a king to put the other king in check, even if after the king is taken by the opposing king, the king doing the taking puts himself into check?
Simply, no. The rules do not allow this.
BUT, there are two occasions where this is allowed:
- Some Chess Variants : Losers Chess for example.
- In Odds Chess. A not so popular odd was to give the weaker opponent's king two squares to move. Thus, it is possible to check the other king.
This is illegal. A king cannot capture another King. The king is the only chess piece that cannot be captured like other pieces, where they are physically removed from the chess board. The kings cannot be on adjacent squares, that is on squares next to each other.
Please note that my response is for normal chess not the other variants where such moves are allowed.