Cross Purposes: an original card game

Cross Purposes is a trick-taking game played with a 52-card pack basically ranking AKQJ1098765432 in each suit. In each deal, one player or partnership chooses a trump suit and the other chooses a 'top rank', thus exercising a variable and contrary dimension of trick-taking power. Whatever rank is chosen as 'tops', the others follow it in descending sequence, turning the corner from Ace to King as necessary. For example, if Eights are tops, then in each suit the ranking order of cards from high to low is 8765432AKQJ109. Though designed for two, it makes an interesting partnership game for four.

Deal: Deal 13 cards each and stack the rest face down. Non-dealer examines his hand and announces either which suit is to be trump (e.g. "Spades"), or which rank is to be top (e.g. "Eights"). If he declares trumps, dealer then responds by declaring tops. Conversely, if he declares tops, dealer responds by declaring trumps.
Play - Phase 1.: Non-dealer leads to the first of 13 tricks. You must follow suit if you can, but may play any card if you can't. The trick is taken by the highest card of the suit led, or by the highest trump if any are played, and the winner of each trick leads to the next. At end of play both players note how many tricks they took and put them to one side.
Play - Phase 2.: The player who did not deal in the first half now deals the previously unused cards one by one till both players have thirteen. Non-dealer examines his hand and announces tops if he declared trumps in Phase 1, or trumps if he declared topsin Phase 1. Dealer then declares either tops or trumps, whichever dimension remains. Tricks are then played as in Phase 1.
Note: You may, if you wish, choose the same trump suit or top rank (or both) as in Phase 1.
Score: At the end of Phase 2 you each score the number of tricks you won in the second deal multiplied by one more than the number you won in the first. For example, if Annie wins 6 in the first and 8 in the second she scores 56 (that is, 6+1 times 8). Benny then necessarily scores 40 (that is, 7+1 times 5).
Game: Play up to 100 points or any other agreed target.
Your questions answered: Andrzej A Lewicki asks: "At the end of Phase 2 you each score the number of tricks you won in the second deal multiplied by one more than the number you won in the first... could you please reveal the mystery behind 'one more than'?"
Answer: It's to reduce the incidence of ties. For example, if you win 6 tricks in the first deal and 7 in the second, and you ignore the extra weighting of one trick in phase one, then you and your opponent will both score 42; but under the actual scoring system you will win by 49 to 48. I could have given the weighting of an additional trick to the second phase, but by this time you reach this you have a pretty good idea of which cards are out of play, so it is easier to judge your choice of trumps or top ranks.

CROSS PURPOSES FOR FOUR
The partnership version

Format: Four may play either crosswise in fixed partnerships or in ad-hoc alliances so that everyone plays for themselves in the long run.
Deal and bid: Deal 13 cards each. There follow one or more rounds of proposals. Each in turn, starting with eldest (dealer's left-hand neighbour), proposes either a trump suit or a top rank. No one may pass. In the alliance game, alliances are established as soon as one player accepts the proposal of another, those two becoming allies against the other two.

Example. West deals, and the bidding goes:
    North: "Spades."
    East: "Threes."
    South: "Jacks."
    West: "Diamonds."
    North: "Threes."

This makes North and East partners with Threes as tops. South and West continue bidding until they agree on a suit.

    South: Clubs.
    West: Hearts.
    South: Hearts.

In the fixed partnership game the same procedure is followed, except that trumps and tops are not established until two players who are sitting opposite each other agree on one dimension, leaving the other two to continue until they agree on the other.
Play: Eldest leads to the first trick. You must follow suit if you can but may play any card if you can't. The trick is taken by the highest card of the suit led, or by the highest trump if any are played, and the winner of each trick leads to the next. Each individual player stores their own won tricks separately from everyone else's.
Score: The score for each partnership, or for each member of an alliance, is 1 more than the number of tricks won by one member, multiplied by 1 more than the number of tricks won by the other.
Example. If partners Annie and Benny win respectively five and one tricks, they score 12 points (i.e. 5 + 1 times 1 + 1), while Connie and Denny, with respectively four and three tricks, score 20 points (i.e. 4 + 1 times 3 + 1).
Game: Play up to 100 points, or any other agreed target.