I think it is likely that the tarot is a cosmograph, a graphic depiction of the structure of the universe. The Tarot of Mantegna is obviously a cosmograph, and it resembles early tarot decks in many points. I assume that the tarot is also a cosmograph, although presumably one representing a different view of the universe than the Mantegna. From that starting point, the question is how to account for the very substantial differences between the two. If the tarot is a cosmograph, it seems it must be of quite a different species than the Mantegna. What is that species?
First, here are the direct correspondences. These are the cards we don't have to explain if we ascribe to the theory that the tarot is a cosmograph on the Mantegna model--
Fool = Beggar
Bagatto = Artisan
Emperor = Emperor
Pope = Pope
Temperance = Temperance
Fortitude = Fortitude
Justice = Justice
Star = Stars ("Eighth Sphere")
Moon = Moon
Sun = Sun
World = Prime Mover
What of the remaining cards? There are some other artistic parallels that have been noticed (Chariot = Mars, for example), but I'm not counting those as support because the placement is usually different enough that these do not automatically support the congruity of the two systems.
I think the remaining cards of the tarot (and the sequence as well) can be explained by the application of just two simple principles:
1. In contrast with the cosmography depicted in the Mantegna, the tarot universe includes allegories of negative forces that constitute a barrier between the world of man and the world of God. There are two subcategories here: (a) forces that make worldly life difficult and unpredictable, and (b) forces that enforce the boundary between the lower world and the higher world.
2. The tarot cosmograph incorporates a heightened interest in pairs of opposites (usually as male/female) operating on the same "vertical" level in the cosmograph.
The presence of Papess and Empress is the first and clearest application of principle 2. With that principle, the "estates of man" portion of the cosmograph is complete.
Moving on to 1(a), the tarot introduces Love, War (Chariot), Fortune (Wheel), and Time (Old Man) to compete with the cardinal virtues in presiding over the affairs of the world. They are all aspects of fate. At least three of these correspond clearly to planetary influences (Love=Venus, War=Mars, Time=Saturn), but they have been dropped down into the sublunar world to show their role in the human condition. So now we have a second tier, which might be called "worldly powers":
Cardinal Virtues: Temperance, Fortitude, Justice
Fates: Love, War, Fortune, Time
In medieval and renaissance cosmography, there is an important boundary between the lower, changing, human world, and the higher, perfect, celestial world. The moon straddles the boundary. Above are the sun, stars, and planets. Below are four elemental spheres: fire, air, water, and earth (in descending order). So the sphere of fire is the edge of the sublunar (lower) world. I think the "dark cards" of the tarot serve to explicate and emphasize this boundary. This is principle 1(b).
The card called Fire, showing a tower with its top caught in a fiery conflagration from above, graphically represents a human attempt to pierce this boundary, and its consequence. The Devil is the ruler of the sublunar world and appears to guard its borders. Death, as the quintessential symbol of the limitation of mortal existence, reinforces the same theme. The Traitor perhaps represents sin (traitor, Judas, criminal). Together, this is a powerful quaternity of forces that prevent humans from ascending into the realm of the immortal God(s).
There are obviously other ways to see these cards. But please note that these "dark cards" immediately precede the "celestial spheres" portion of the tarot sequence (stars - moon - sun), exactly where the boundary of the mortal world would be.
The Angel is cosmographically sound because the spheres of the angels are between the celestial spheres and God in the cosmographies of the time.
Is the World card "God"? The writer of the Steele sermon, writen some time in the 15th century, says so, referring to the card as "The World (that is, God the Father)".
There are some points glossed over, of course. Why does The Star come before the Moon and Sun, when the fixed stars were universally recognized to be beyond the spheres of the planets? I won't claim to know "the answer", but an appeal to principle 2 may help. The Moon and Sun are the quintessential male/female pair. Since the tarot cosmograph does not make room for each separate planetary sphere, as the Mantegna series does, it could be that the celestial spheres were collapsed into The Stars, signalling the entry into the celestial world, followed by the Moon and Sun as the dual prototype of the heavenly realm.
So now, without appealing to much except the Mantegna cosmograph and two principles of difference, we have put together the Tarot cosmograph:
Estates of Man: Fool, Bagatto, Emperor and Empress, Papess and Pope
Fates and Virtues: Love, Chariot (War), Wheel (Fortune), Old Man (Time); Temperance, Justice, Fortitude
Traitor, Death, Devil, Fire
Stars, Moon, Sun, Angel, World (God)
I haven't gotten into the exact placement of the sublunar cards. That is a "higher order" project. There are two possible difficulties, both having to do with a virtue being "promoted" to a higher level in the cosmograph. Justice is sometimes placed next to the Angel. That is still cosmographically plausible if Justice is taken as the archangel Michael rather than a cardinal virtue. Both understandings of the figure bearing sword and scales were common at the time. The other case involves the promotion of Temperance (in Tarot of Marseilles) into the boundary region. I feel this has something to do with the fact that in the Marseilles, the virtues are placed exactly three postions apart. I suspect this may be yet another reflection of principle 2 (dualism): the Marseilles designers saw the virtues (in proper Aristotelian and scholastic fashion) as reconcilers of dual opposites, and so we get Lovers (Venus/female) - Chariot (Mars/male) - Justice (balance of opposites). Admittedly, Old Man / Wheel and Traitor / Death are not so obvious as dual pairs, but it still may be that feeling each virtue should be preceded by a pair may be what elevated Temperance to XIV. It's also interesting to note that Temperance sports angel wings much more often than the other virtues, and only in Tarot of Marseilles and its derivatives.
That's a bit of a sidetrack. (Trust me, I've been down many of them!) The bottom line is that I think it is quite legitimate to view the tarot as a cosmograph, and in particular a cosmograph distinguished by its emphasis on duality and the assertion of a frightening barrier interposed between man and God.
What does that mean? What are we dealing with? I don't have the historical acumen to offer conjectures about religious/philosophical sects that might have been bouncing around in northern Italy in the early 1400s that might have created or endorsed such a cosmograph. There is, however, a convenient label for this combination of ideas: gnostic. I'm not implying that anyone in the 15th century would have used this word to describe the tarot, but sitting here in the 20th century, it seems to me a fair adjective.
Final Report of this field taxonomist--
Specimen: Tarot Trumps
Genus: cosmograph
Species: gnostic