There are several stories regarding this symbol, but you will often see it symbolizing Okinawa. You will see it on all things from Okinawan products, Okinawan organizational logos, to Okinawan martial arts patches. Here are a few of those legends regarding the origin of this symbol:
"It is believed that three envoys were sent to the Shogun of Japan to negotiate a treaty between the mighty Ryukyu and the mainlanders. The 3 Ryukyuan envoys were ambushed and taken before the Shogun. One of the Ryukyuan men was Zanna (Janna) Oyakata, the King's scholar, he attempted to reason with the mainlanders but he and the other 2 Ryukyuans were killed. The method of death was particularly heinous, they were boiled to death. The Mon (or the Ryukyuan King's Mon, very similar to this one) represents the 3 bodies swirling in the water lest we forget their sacrifice."
"In the time of king Sho Nei, a small group of people loyal to the Okinawan king rebelled against the Satsuma. They were sadly captured and ordered to their death and forced to jump into a large pot/vat of boiling water. As they met their unfortunate deaths with their own defiant dignity, a swirling pattern developed in the boiling water. It is said that this pattern inspired those watching to adopt the symbol now called "Hidari Gomon/Hijai Gumun". This symbol closely resembles the Japanese "mitsu domae" symbol."
Another possibility is that the symbol in mention is also said to represent the three kingdoms of the Ryukyu's (sanzan) mainly: Hokuzan, Chuzan and Nanzan. One of the younger kings of the Sho dynasty, Sho Taku, adopted the Japanese God of War symbol (hachiman) or mitsu domoe. Often, in karate-do circles, this symbol (mon) is used as a crest for that particular style and is often referred to simply as the "Sho" in what is a reference to the Sho dynasty.
"It is believed that three envoys were sent to the Shogun of Japan to negotiate a treaty between the mighty Ryukyu and the mainlanders. The 3 Ryukyuan envoys were ambushed and taken before the Shogun. One of the Ryukyuan men was Zanna (Janna) Oyakata, the King's scholar, he attempted to reason with the mainlanders but he and the other 2 Ryukyuans were killed. The method of death was particularly heinous, they were boiled to death. The Mon (or the Ryukyuan King's Mon, very similar to this one) represents the 3 bodies swirling in the water lest we forget their sacrifice."
"In the time of king Sho Nei, a small group of people loyal to the Okinawan king rebelled against the Satsuma. They were sadly captured and ordered to their death and forced to jump into a large pot/vat of boiling water. As they met their unfortunate deaths with their own defiant dignity, a swirling pattern developed in the boiling water. It is said that this pattern inspired those watching to adopt the symbol now called "Hidari Gomon/Hijai Gumun". This symbol closely resembles the Japanese "mitsu domae" symbol."
Another possibility is that the symbol in mention is also said to represent the three kingdoms of the Ryukyu's (sanzan) mainly: Hokuzan, Chuzan and Nanzan. One of the younger kings of the Sho dynasty, Sho Taku, adopted the Japanese God of War symbol (hachiman) or mitsu domoe. Often, in karate-do circles, this symbol (mon) is used as a crest for that particular style and is often referred to simply as the "Sho" in what is a reference to the Sho dynasty.