So I see these two martial arts/ceremonies mentioned as an origin of both sumo and aki-jutsu. But they are also sometimes refrenced as being more similar to greek pankration or at least a more mixed style of both striking as well as grappling. So I was wondering if we have any evidence or an educated guess what they would look like.
Sumai no sechie 相撲節会 is the "origin" of term sumo, coming from sumau which is one of many words meaning "to fight" or "exchange of blows." Not sure when the pronunciation of "sumo" comes in vs "sumai" but dictionaries like the Nippo Jisho 日葡辞書 (Japanese to Portuguese dictionary) from early Edo pronounce 相撲 as "sumo." As far as I'm aware, we don't know exactly what kind of techniques were used back in the days of the Kojiki and Nihon Shoki (early 8th century); a lot of codification of martial art techniques in things like densho (傳書 "transmission writings") and makimono (巻物 scrolls though literally "rolled things") until probably the 13th or 14th centuries. Early systemization of fighting in Japan is a bit complicated with lots of debate taking place amongst practitioners and historians. For example, Dr. Karl Friday writes in Legacies of the Sword about people being aware of sword skill in the 8th and 9th centuries, but we don't know what kind of teachings existed specifically. This is in part due to the age (before a common writing method existed; Chinese was for the elite/aristocracy) and also due to the culture of secrecy that existed back then (I discuss a little in this thread). A few stories of people being told to go teach soldiers and stories of learning from tengu and spirits, but those are stories which may or may not be based in truth.
However, it's safe to say the methods used back then were not like the sumo we see today. Outside of the ritual aspect of sumai no sechie festivals (which shouldn't be ignored), the original methods used were for actual fighting rather than sport. In the Nihon Shoki there's a record of a match between Nomi no Sukune 野見宿禰 and Taima no Kehaya 当麻蹴速 after both claimed to be the strongest men of Izumo and Yamato provinces, respectively. Nomi no Sukune wins with kicks, crushing Kehaya's ribs, hips, and killing him. Nomi goes on to become a guard in the imperial palace. This is not surprising as Joichi Nagano in his work Konjaku Monogatarishu no Kansho to Hihyo 今昔物語集の鑑賞と批評 (commentary/review of the Konjaku Monogatarishu) points out how the Hyobusho (兵部省, the Ministry of Military Affairs established under the ritsuryo system) helped recruit and supervise fighters from within the Konoefu 近衛府- one of the palace guard groups at the time (there were a lot guard variations in the 8th and 9th centuries).
Donn Draeger and Robert Smith in Contemporary Asian Fighting Arts back in 1969 have this to say about sumai-
Fully effective in combat, the object of this ancient fighting was to cause on of the opponents to surrender unconditionally. Killing was permitted. As a fighting system, early sumo depended heavily upon close-quarter grappling much like modern sport, but differed widely from the present version in that the range of techniques included kicking, butting, and striking. A stamp-and-drive type of kicking differing from the whiplash type of karate styles today was used. It aimed at knocking the opponent to the ground where he could be trampled into submission. Sumo at this time was not a general skill possessed by the warriors for use in mass combat situations. It was used only by chosen fighters representing various sides in a dispute.
Of note is that Emperor Ninmyo in the mid-9th century felt that sumai was to be regarded as a symbol of military strength. Techniques and methods were recognized. Draeger and Smith feel that the military focus on combat led to sumai becoming kumiuchi (組打, generally meaning "grappling or wrestling methods"... but not really...). Sumai as sumo would morph into a popular sport for festivals where more dangerous techniques like "striking with fists" would become prohibited. It fell out of popularity in the 11th-12th centuries as the bushi class emerged into power. Given sumai's theoretical origins in Chinese techniques and other teachings that warrior families like the Otomo and Mononobe were theoretically responsible for, it's not surprising that "fake fighting" was displaced in favor of other concepts.
As for specific techniques, the Konjaku Monogatarishu references a few in its stories, including punches, kicks, grabbing to the throat and armpit, grabbing the belt, hooking the legs with what might now be described as uchigake 内掛け or sotogake 外掛け, and attacks to the knees. In the Kakutogi no Rekishi 格闘技の歴史 (book on the history of unarmed fighting arts), a section on sumai references usage of kake 掛け (a kind of "hooking"), nage 投げ (throwing), hineri 捻り (a "twisting"), shime 絞め (chokes/constrictions- blood, air, and pain), uchi 打ち (striking), tsuki 突き (thrusts/weak point attacks), keri 蹴り (kicking), hazushi 外し (like dodging/evading), kansetsu waza 関節技 (joint techniques), and kobushi uchi gata 拳打型 (forms related to striking with fists).
Unfortunately I don't have a copy of Ichiro Nitta's book Sumo: Sono Rekishi to Giho 相撲 その歴史と技法 (history and techinques of sumo) to see what else might be listed there. It has been ordered for delivery in a couple weeks though!
Tegoi 手乞 is something I have very little knowledge of. Various sources allude to it being at the origin of sumai and other things, especially in Daito Ryu aikijutsu. If it's that old, I'm very confident in saying tegoi is not about technique but rather concepts. After all, the word literally means "hand (te) invite (gou)." This is where knowledge of abstraction in Japanese culture comes in. Te 手 means "hand" but in the olden days, it meant technique. Thus the idea of tegoi is most likely likened to the notion of divine techniques or techniques that were received in some capacity (tengu, divine inspiration, meditations, etc). Again, the lack of written records at the time coupled with lack of codification of martial art teachings makes it very difficult in my opinion to say what tegoi really was or what it consisted of. It's like asking what gave birth to the technique of punching someone in the face or when did people learn to do block-punch combos.
Given the close association of tegoi with Daito Ryu (whose history is complicated prior to Takeda Sokaku in the 19th century), this is what aikido scholar Stanley Pranin had to say about it in the old Aiki News journal-
Daito-ryu Aiki Budo has its origins based on the concept of "tegoi". The concept of tegoi comes from a passage in the Kojiki which reads as follows: "... When Takeminakata no kami took the hand of Takemikazuchi no kami, the hand changed into a column of ice, then again changed into a sword blade and the latter was completely hopeless. Then Takemikazuchi no kami in turn took the hand of Takeminakata no kami. He held it as if grasping a young reed and cast it aside". It is this tegoi which is said to be at the origin of Sumo and its techniques have been transmitted as "Aiki In-Yo (yin-yang) method" and now constitute the basic techniques of Daito-ryu Aiki Nage. This form of tegoi was transmitted for a long period until the Sumo Assembly of the Kamakura warriors (Kamakura Period = 1192-1333). In 868 during the reign of the Emperor Seiwa, supervision over court banquet Sumo was transferred over to the Board of Military Affairs and was controlled by the Imperial guards thereafter. From that time Sumo developed as a martial technique. The art continued to be transmitted through Emperor Seiwa's descendants of the Seiwa-Genji family for six generations through Shinra Saburo Minamoto no (Genji) Yoshimitsu.
And this is what the Daito Ryu website says-
Aiki is said to have originated in the ancient art of tegoi, which is mentioned in an ancient Japanese myth about two gods, Takemikazuchi no Kami and Takeminakata no Kami. Recorded in Japan's oldest written document, the Kojiki, (Records of Ancient Matters, compiled around 712 AD), this story recounts how Takemikazuchi no Kami took the hands of Takeminakata no Kami and "as if he had taken hold of a reed, squeezed his hands and threw him."
Tegoi is also said to be the origin of sumo (now Japan's national sport), recounted in the legend of Nomi no Sukune and Taima no Kehaya in Nihon Shoki (Chronicles of Japan, 720 AD). During the Heian era (c. 792-1192) tegoi was transformed into a court entertainment known as sumai no sechie, which in turn subsequently evolved into sumo, practiced by warriors of the Kamakura era (1192-1333). Sumai no sechie was a sumo competition during which wrestlers from all over Japan competed before the Japanese emperor. Unlike modern sumo, however, there was no wrestling ring, and the techniques used were more combative than those used in sumo today. The combative character of the art at that point of time is clearly seen even from an imperial edict issued by emperor Nimmyo (810-850), saying that "sumai no sechie is not just an entertainment; it is an ideal means for cultivating real martial skills."
Hopefully all this helps a little. While martial art history is a passion of mine, I learned long ago that a lot of concepts remain the same and that to try and figure out what a technique was like 1,000+ years ago isn't really worth it. The fighting techniques and concepts that prevailed lived through to today.... along with a lot of Edo and Meiji-era profiteering schools that kind of suck. Better to understand the techniques of the day through training in armor/weapons of the period than guessing based on modern budo IMO.
Sources-