Sorry if this is not the right sub for this question.
The only swordsmen in Japanese history that's famous for dual wielding that I know of was Miyamoto Musashi. I'm not sure that he held his sword reverse grip on his left and normal grip on his right. Are there any historical background on this swordstyle? Or is it just video game, anime, etc making it up because it looks cool. Here are some picture for references
It isn't a historical Japanese style. Traditional Japanese two-sword styles have survived to the present, and the kata in these schools show the techniques:
Two sword styles were used outside Japan. There are surviving Chinese two-sword forms (generally solo forms, rather than the two-person kata typical of Japan, which makes them harder to decipher):
Korean two-sword style has been reconstructed from the military manual Muyedobotongji:
and also survives in traditional sword dances:
Two swords are used in Thai martial arts:
and were also used in India.
Some European martial arts manuals cover twin rapier:
https://wiktenauer.com/wiki/File:Sutor_74.jpg from https://wiktenauer.com/wiki/Jakob_Sutor_von_Baden
https://wiktenauer.com/wiki/File:Hundt_081.jpg from https://wiktenauer.com/wiki/Michael_Hundt
and it sees some use in modern reconstructions of historical European martial arts (HEMA):
Note that all of these use a conventional forward grip for both swords, rather than reverse grip. Reverse grip is rare for swords historically, because for most purposes it is inferior to a conventional grip. It has seen some use when stabbing with a sword icepick-style:
but this is of little use when using two swords. With two swords, using one (or both) in reverse grip makes the reverse grip sword much less dangerous offensively, and also less use defensively. It also makes it harder to use the two swords together usefully. As I noted in https://www.reddit.com/r/AskHistorians/comments/gbckta/why_were_quillonscrossguards_not_popularized_in/ a two-handed spear is a very dangerous weapon to face with just a sword: after the sword is used to parry a spear thrust, the spearman can quickly disengage, in a haf or three-quarter circle around the hilt, and attack again. Very long quillons are one way to make this disengagement by a spear harder. Perhaps even more effective is to use two swords, held with the blades crossed to parry the spear (you can see the two swords crossed like this at times in the Japanese and Chinese kata/forms).
That said, some styles of Chinese martial arts, notably some branches of Wing Chun, some branches of Hung Ga, and some Shaolin styles, will use two swords (butterfly swords) with one of the swords sometimes held in reverse grip:
This use of the reverse grip was reported in the mid-19th century (Bingham, 1843, pp 131-132):
These twin swords, when in the scabbard, appear as one thick clumsy weapon, about two feet in length; the guard for the hand continuing straight, rather beyond the "fort" of the sword turns towards the point, forming a hook about two inches long. When in use, the thumb of each hand is passed under this hook, on which the sword hangs, until a twist of the wrist brings the gripe within the grasp of the swordsman.
where the thumb under the hook on the guard must be in reverse grip. Notably, the hook is similar to the hooks on the sai, which is also used with reverse grip, for thrusting strikes with the pommel, and for blocking with the weapon along the forearm:
The sai, today used in Japanese karate, was used in Okinawa as a police weapon, and previously came from China where it was also used as a police weapon. The reverse grip butterfly swords might be adapted Chinese sai technique, but it isn't possible to tell. What one can say is that one of the major disadvantages of reverse grip - giving up reach - matters less if there is less reach to be given up. When both weapons are quite short, and one has a large reach disadvantage against longer swords, a slightly bigger reach disadvantage doesn't make things that much worse. If fighting with two swords against a sword of similar length, then giving up reach by using reverse grip has a much larger disadvantageous effect.
So, in summary, there was very little historical use of this forward + reverse two-sword style. Two-sword styles are relatively rare, compared to single sword and sword + shield, but many examples can be found fairly readily. Among them, it is only some branches of southern Chinese martial arts that use reverse grip, some of the time, with short butterfly swords.
Reference:
J. Elliot Bingham, Narrative of the expedition to China, from the commencement of the war to its termination in 1842. Vol. I, 2nd ed., Henry Colburn, London, 1843.
