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Rules: 

Ordinary rules of chess apply with the major exception that each player is
granted the ability to move twice but only once per game. If both players
have used their double move, then standard chess rules apply without
exception.

In the discussion here, to announce checkmate means that the opponent king
cannot avoid capture without losing the double move. The strategy is to
force the opponent to use the double move. It is an extreme disadvantage
to not have this when the other player does. It is worth the sacrifice of
a Queen plus Rook plus Bishop (at least). 

In response to 'checkmate', the player could simply use the double move
and thus there was no 'checkmate' after all – but that is nearly always
short-lived satisfaction. Mate is almost immediate afterward.

Giving check by threatening to take the double move does not count as
actually taking it. You can threaten all you want.

Special Rule #1.

You cannot move your knights until after you have lost at least one piece
or pawn. Why? Because otherwise White advances knights immediately and
checkmates Black every game.

Special Rule #2.

Assume that in regular chess there were no 'check' concept. Instead, one
simply takes the opponent King. But assume further that if one lost the
King, one could take the last move back without penalty and try some other
move. That, in effect, is what 'check' does. It simply avoids the
indignity of taking back moves, etc.

Well, using that notion of the meaning of 'check' and applying the
double move, one can derive additional rules.

a) If you have the double move but your opponent does not, you can even
mate with the King! You threaten to take one move placing your King
adjacent to the opponent and another move, taking the opponent King.

b) If both players have the double move, neither King can move within two
squares of the other King.

c) If you have a pawn on the seventh rank you win if your opponent's King
cannot avoid capture for all possible promotion cases. Example: You 
threaten to promote to a Queen and then take the King with the Queen. But
if your opponent moves to a square safe from that threat, you would
promote to a Knight and thus take the King anyway. There is no escape
(other than using the double move which is doom anyway.)

Note:

There is no special rule regarding en passent captures. Using standard
rules, a player could elect to advance pawn two then one more and thereby
avoid en passent capture. However, nobody in their right mind would ever
sacrifice the double move just for that purpose.

History: Invented by QBasicMac in 1970 and played by me and three others
until our group broke up in 1973. Really fun when Kibitzers are present to
be confused by seemingly random calls of 'check'.