So, I had an idea for a game using the Decktet. Seeing as most of the cards have multiple suits, the following might work.
The game would be a "tree-building" game, which much like tableau builders, would involve players taking turns building piles of cards. The difference between a tableau builder and a tree-builder, is that the trees grow into multiple piles over time.
So, to start the game, take the extended Decktet, and take out the six Aces. Shuffle those and place them face down in the center of the table. These are your "root" cards. Turn the top two over, and move them to spacious areas of the table. These are the starter trees.
Shuffle the rest of the deck together, and deal 5 cards to each player to form the hand.
Gameplay is in turns, each turn a player either plays a card building on a tree, harvests a tree, or unroots a tree. Regardless of what they do, they draw a card at the end of the turn. There is no maximum hand size.
Playing a card on a tree involves playing a card which shares a suit and a higher rank than the card that you are playing on. All cards aside from Crowns may be played off of, and all aside from Pawns and Courts may have two cards played off of them. Pawns and Courts may have three cards played off of them. For the purposes of this game, Pawns and Courts are considered of equal rank, and their rank lies between Ace and Two. Crowns may not have any cards played off of them, and as such form ends of branches.
Harvesting a tree is how you score points. A player scores points for the number of Crowns in a tree they harvest. A tree with zero Crowns may not be harvested. When they harvest a tree, they collect the Ace and place it in front of them, then return the rest of the cards to the discard pile. When harvesting future trees, they gain bonus points if Crowns in those trees share suits with any aces they own.
If a player cannot play a card, nor can they harvest a tree, they must unroot a tree. Additionally, unrooting cannot be a voluntary action. Unrooting a tree involves shuffling that tree's Ace into the root pile, returning all other cards to the discard pile, and starting a new tree (see below).
Starting a new tree is done by shuffling the root pile and flipping the top card over to form a new root. This is not an action and does not take a turn. It is mandatory whenever there are less than two trees in play, and the root pile is not yet exhausted.
When a player must draw and there are no cards to draw, the discard pile is reshuffled to form a new draw pile.
The game is over when all six roots have been harvested. The player with the most points wins.
I haven't played it yet, but it could be fun. I'm sorry if the idea is clear as mud at the moment, but this is more of a ramble than a fully fleshed out ruleset.