Though most calendar systems divide the year into 12 or 13 months, each being about 4 weeks in length, it would make just as much if not more sense to divide it instead into 4 seasons of 13 weeks each. This would add up to 52 weeks, or 364 days, so to keep the seasons aligned, a leap week needs to be added about once every 5 years.
If we want to align the seasons such that they start close to the solstices and equinoxes, we can use the fact that the time between the June solstice and September equinox is about 3.7 days longer than between the December solstice and March equinox, so in my calendar I've decided to put the leap week at the end of the "summer" season (for the Northern Hemisphere), and to have each year begin in what we call September.
Now, I could just number the weeks of each month from 1 to 13, but that's boring. Instead, the three weeks after the 10th will be called Jack, Queen, and King. Oh, and the first week will be named Ace for good measure.
It might be nice to have a special name and symbol for the seasons as well, so perhaps:
Fall is the Spades Season (♠), to symbolize the need to dig up fallen leaves
Winter is the Diamonds Season (♦), to symbolize the crystalline ice and snow
Spring is the Clubs Season (♣), to symbolize growing plants and mosses
Summer is the Hearts Season (♥), to symbolize the warmth of love
These work for the Northern Hemisphere, but for the south:
Spring is the Spades Season (♠), to symbolize digging the soil in preparation for crops
Summer is the Diamonds Season (♦), to symbolize the shining of the sun
Fall is the Clubs Season (♣), to symbolize the look of the leaves that fall to the ground
Winter is the Hearts Season (♥), to symbolize the warmth of love, needed when it's dark and cold
These four seasons, and the occasional leap week, which I've decided to call the Joker week, come together to form a Deck, having either 364 or 371 days in total. 22 of these decks compose a Shoe, such that a Joker is placed into 4 selected decks spread throughout the shoe. The names of these decks are based on the Major Arcana, with the current deck, lasting from 2023–2024, called the Deck of the Fool. Jokers are included in decks of the Fool, the Hierophant, Justice, and the Tower, with an additional rule that if the Joker for a deck of the Fool were to occur in a gregorian year that ends in a 0, that Joker is skipped. Equivalently, there is a 220 year long cycle consisting of 10 shoes, where in the first shoe of the cycle (in this case the one that began in 1979), the Joker is dropped for its deck of the Fool. These rules keep the seasonal drift to about one day every 800 years.