Personally, I like to focus instead on the Bologna tarots, as in the rules there is a special relationship between the Fool and the Magician (as explained to me a long time ago by Ross... Ross, can you describe it?). I also wonder about the Bologna because of the traditional lack of numbering. I'm under the impression that in the Bologna rules, there is a special relationship between the Fool and the Magician, and another special relationship between the "Papi", the next four trumps. So in that grouping, we have rules to support the idea that there is a relationship between these cards beyond just the iconographic. But so often, when I read about tarot, the Fool is considered *outside* of the trumps, and I think it might be harder within that frameset to place the Fool into a group at the start of the sequence.
The Fool is an interesting card, and as a wildcard (as he can be in Bolognese games) or a one-off substitute (as he is in Bologna and most other places), his role is always unique among the added cards. Since wildcards have been invented independently in other card games - the Chinese "money packs" have one, and of course the Joker came later, in the 19th century - and, the Jack of whatever suit was sometimes given special powers, like in Karnöffel where he is highest of the "chosen" suit (equivalent to a suit of partial trumps), or the Juker in Euchre, which is probably the origin of the Joker in American card decks - it might be theorized that the Fool existed independently as a wildcard or substitute card before the full trump sequence we know was invented. There is no specific evidence for this, but it is good to remember that Tarot also preserves the 4-court deck (56 card) deck, that was apparently well-known in Italy in the early 15th century, which disappeared in all the games we know that are played with standard decks in Italy (I don't think there is a kind of Tarot anywhere that uses only 3 court cards, nor a regular deck that has 4 court cards (have to check the Hofjagspiel, mid-15th, it might have it, but it is not standard)), so, if a theory needs it, it is not implausible to suggest that other such "fossils" might have become subsumed in the standard trump sequence. Fernando de la Torre's Spanish deck, created around 1450, only has one trump, an Emperor. From what he describes, it was only a trump, a card which beat all others, not a wildcard that could substitute for another in a combination. But what is interesting about Fernando de la Torre is that he spent at least a couple of years in Florence in the early 1430s, so it is natural to wonder, since he was a card game inventor, whether he knew Tarot then? If it existed then, and he saw it, why didn't he make a game with more trumps? Or, is it possible that in Florence at the time, some kind of game existed with only one trump, perhaps an Emperor?
We can't know, but there is a basis for speculation.
Taking it the other way, that all of the special cards in Tarot (excluding the Queens, which did pre-exist) were invented at once, then your question again demands that the interpreter make explicit his various paths of reasoning.
One important premise is whether the trump sequence can be interpreted on its own, as a sequence of images not necessarily part of a card game, or whether the order and icongraphy reflects some rules of the play itself, and can only be properly interpreted by reference to those rules. Michael and I differ on this fundamental point, he representing the former and I the latter opinion.
On the one hand, with Michael, we can note that the Fool makes an obvious pair with the Bagatto - at least in the earliest types of A and B, where he is a performer. This is not so obvious in the Visconti and C types, where he might be better interpreted as a beggar or vagabond, rather than a magicians's assistant. So is he supposed to be a professional fool, or is he really a madman? The answer might affect our interpretation of the sequence, since it seems to have affected the iconography of the various early decks. We can note in general though that magicians doing table-top tricks in many Children of the Planets illustrations of the Children of the Moon are accompanied by assisstants whose role is no doubt to soften up the crowd, by playing instruments, telling jokes, juggling, and whatever else is necessary to help get the crowd to the table.
In any case, the common point between insane vagabonds and professional fools is that they act crazy, and that they are entertaining. The Bagatto is an actual entertainer in all cases, and is not insane. Hence Michael's explanation that this pair represents "fools and conmen" in general.
On the other hand, we can note that in all (but one, the "Belgian Tarot" where he is numbered XXII) Tarot games known the Fool is unnumbered and plays the special roles noted above. He is not fixed in the sequence as it is played. This suggests why the image of a Fool may have been chosen for this role, since Fools move around freely in a crowd and do what they want, outside of the normal bounds of decorum and order. So we can distinguish him from the Bagatto, who always has a fixed place and must obey the rules. They do not form a natural "ludic" pair.
But, to address your question about Bologna, he does form a ludic pair of a sort in the classic form of this game, where both cards are wildcards - they can substitute for others in a combination (for which reason they are called "contatori" - counters, since they "count" for others), as long as they themselves are not sequential in the substitution. And, the Bagatto (Bégato in Bologna) has no number. The papi, unnumbered, are four, but Love is 5, which means that the Begato has no number (as he does not, but even with the numbered decks, the numbering starts at 5). But as is the universal habit, only the Fool can be used as "excuse", a chance to save a valuable card (like a high trump or a king) that would otherwise have to be played.
So, in Bologna both are unnumbered, but the place
of the Begato is fixed - he is the lowest trump. He has ordinal but not cardinal value in this form of deck.
In the A types from Florence and Bologna (and the Este deck), the Fool can obviously be taken as the complement of the Bagatto, since he is a performer, not a vagrant.
I'm not sure if this has helped sort out the problem for you (since I haven't addressed the papi
). If I go with the theory that all of the trumps were invented at once, it seems most probable that he forms a pair with the Bagatto, I think, but one which reflects their actual roles in society. If he is a wildcard from a former kind of game that survived into Tarot, then it could be theorized that he inspired
the Bagatto, since these professionals were accompanied by such figures.