Waite does, and most others...
each minor arcana suit of earth (pentegram/coin), air (sword/atheme), fire (wands/clubs), water (chalice, vessle) has a king, queen, knight & page or their variation. each correspondence also decribes a direction and a color, and mental state. Here is a list of "correspondents" which I have gathered from a variety of sources. I am sure they are not all that exist (most are strictly from "Westren/European history," some might be disputed, but I feel it is a fairly accurate list). You will see these things show up on the cards. Hope this is helpful.
Air:
Direction: East
Season: Spring
Point in life: Childhood/ Youth
Time of Day: Sunrise
Elementals: Sylphs
Color: Yellow or Pale Blue
Magical Virtue: To Know
Magical Tools: Atheme, Sword, Censor
Archangel: Raphael
Jewel: Quartz, diamonds, and Topaz
Metal: ?
Astrological Rulers: Mercury
Wind of the East: Eurus
Zodiac Signs: Gemini, Libra, Aquarius
Tarot Suit: Swords
Essential Oils: Lavender, Lemon, Violet, Rosemary, Frankincense
Creatures: Birds and winged creatures
Plants: Lavender, Marjoram, Mint, Caraway
Trees: Hazel, Aspen
Images: Knives, Swords, Arrows, Science, Analysis, Mountains, feathers, mirrors, wind instruments, perfume, poets, singers, teachers, books, scrolls, computers
To contact: Wind Chimes, bells, whistles, flutes, walk in the wind, analyze sound, breathing exercises, IQ tests, mental puzzles
Goddesses: Arianrhod, Athena, Minerva, Nuit
Gods: Enlil, Mercury, Hermes, Thoth
Air is the focus of the Spring Equinox, Eostre, and Beltain (with Fire)
Fire:
Direction: South
Season: Summer
Point in life: Prime of life
Time of Day: Midday
Elementals: Salamanders
Color: Red, Orange, Flame colors
Magical Virtue: To Will
Magical Tools: Wand, Candles
Archangel: Micheal
Jewel: Quartz, fire opal, red Jasper, ruby, bloodstone, garnet, lava, carnelian, tiger eye, agate
Metal: Gold and Brass
Astrological Rulers: Sun, Mars, Jupiter
Wind of the South: Notus
Zodiac Signs: Aries, Leo, Sagittarius
Tarot Suit: Wands
Essential Oils: Pine, Basil, Calendula
Creatures: Lions, Dragons, Lizards, Snakes
Plants: Heliotrope, Sunflowers, Basil, Chilies, Nettles
Trees: Oak, Ash
Images: Life Blood, Flames, Bonfires, Candles, Hearth, Lamps Solar Glyphs, Phallus, Divine King, Deserts, Volcanoes, Staffs and Staves, Infernos, Heart, Courage, illumination, Power
To contact: Wind Chimes, Fire, sunbathe, Sauna, Bikram Yoga, etc.: Get hot
Goddesses: Brigid, Freya, Hestia, Vesta
Gods: Ra, Vulcan, Apollo, Baldur, Bel, Horus
Fire is the focus of The Summer Solstice and of Imbloc with Water, Beltain with Air, Lammas with Water, Samhain with Earth
Water:
Direction: West
Season: Autumn
Point in life: Middle age
Time of Day: Evening/ Twilight
Elemental: Undines
Color: Sea Green, Blue, Aqua
Magical Virtue: To Dare
Magical Tools: Cauldron, Chalice
Archangel: Raphael?
Jewel: Aqua, pearl
Metal: Silver, Copper, Quicksilver
Astrological Rulers: Venus, Moon
Wind of the West: Zephyrus
Zodiac Signs: Cancer, Scorpio, Pisces
Tarot Suit: Cups
Essential Oils: Jasmine, Rose, Vanillas, Tuber rose, Lilly
Creatures: Sea Creatures, Citations, Sea Snakes
Plants: Apple, Rose, Elder, Blackberry
Trees: Alder Willow
Images: Pools, water ways etc., Mothers, nurses, careers, translucent crystals, movement, mist, receptiveness, dark mirrors, holy wells, womb, menstruation, cleansing, rain, storm ice, flood
To contact: Bathe, beach, swim, cry, laugh out loud, be emotional, eat "emotional food"
Goddesses: Aphrodite, Tiamat, Isis
Gods: Manannan, Poseidon, Neptune
Water is the focus of the Autumn Equinox, of Imbloc with Fire, of lammas with fire
Earth:
Direction: North
Season: Winter
Point in life: Old age
Time of Day: midnight
Elementals: Gnomes, Dwarves
Color: black, Brown, earth tones
Magical Virtue: To Be Silent
Magical Tools: Pentacle, Shield
Archangel: Auriel
Jewel: Stones in particular; obsidian, Jet, and dark crystals
Metal: Iron
Astrological Rulers: Saturn, Earth
Wind of the North: Boreus
Zodiac Signs: Taurus, Virgo, Capricorn
Tarot Suit: Pentacles
Essential Oils: Patchouli, Vetiver, Sage, Cypress
Creatures: Stags, Serpents, Wolves, Bears
Plants: Poppy, Fern, Barley, Corn
Trees: Beech, Cyprus
Images: Caves, Groves, Standing Stones, Forests, Orchards, Tombs, Burrows, Feasting, Long Barrows, Pyramids
To contact: Visit places above, walk in the woods, mazes and labyrinths. Study Pottery, Archeology, Geology
Goddesses: Gaia, Demeter, Ceres, Persephone, Flora, Cerridwen, Blodeuwedd
Gods: Pan, Herne, Cernunnos, Dumuzi
Earth is the focus of The winter Solstice and Samhain with fire
