The ‘Ryōi Shintō-ryū Denju Mokuroko’ (良移心当流 伝授目録) believed to have been writen by Fukuno Shichirouemon (福野七郎右衛門), explains the meaning of 'Ryōi Shintō Wa / Yawara' as follows:

“The meaning of 'Ryōi Shintō Wa / Yawara' is to shift (i 移) your heart (shin 心) well (ryō 良), then you will encounter (tō 当) harmony (wa 和, or yawara, an alternative tern for jujutsu).”

Alternatively, the ‘Ryōi Shintō-ryū Ryakushi’ (良移心当流略史 -A brief history of Ryōi Shintō-ryū) records Fukuno as saying that: “Ryōi Shintō means to: skilfully apply the mind (shin 心) and strike (tō 当) with ease”.

It may be that Ryōi Shintō-ryū founder - Fukuno Shichirouemon was influenced, or inspired by the name of one of the older 'Seven (sword) schools of Kantō / Kashima' called 'Ryōi-ryū' (良移流 - good / skilled, shift / move style). The term 'Shintō' was also a popular title for schools and various kanji were used

The 'Ryōi-ryū' is mentioned in the article: 'Some Thoughts on the Emergence and ‘Aesthetic Asceticism of Ryūha-bugei' by Alexander Bennett
(Publ: Kokushikan University Institute of Budo and Moral Education "Butoku Kiyo Butoku Kiyo" No. 26.)

Under the heading of: 'Table outlining Japan’s first combat ryūha-bugei:' Bennet details the "The ‘7 Schools of Kantō (関東七流)" and states that:
"This classification of schools was considered by scholars from the Tokugawa period to represent the main streams or branches that evolved in the eastern provinces.


1. Kashima (鹿島)
2. Katori (香取)
3. Honshin-ryū (本心流)
4. Bokuden-ryū (卜伝流)
5. Shintō-ryū (神刀流)
6. Yamato-ryū (日本流)
7. Ryōi-ryū (良移流)”

(Reference:https://researchmap.jp/read0084366/published_papers/22375150)

This is corroborated by the (Japanese language) 'Tenshinshō Jigen-ryū Kaisetsu (天眞正自源流解説) Shōbu no Kan (尚武之間)' website, which states that:

“Kashima's swords came to be called Jōko-ryū (上古流 - upper ancient style) and Chūko-ryū (中古流 ancient style) and were handed down mainly by the Urabe (卜部家) and Kikkawa clan (吉川家), who are the priests, and the seven families with excellent work were called Kantō nana-ryu (Kantō i.e. north-eastern half of Japan seven styles), or Kashima nana-ryu (鹿島七流 - Kashima seven styles). became the seven schools of

• Ryoi-ryū (良移流),
• Kashima-ryū (鹿島流)
• Katori-ryū (香取流)
• Honshin-ryū (本心流)
• Bokuden-ryū (ト伝流)
• Shinto-ryū (神刀流)
• Yamato-ryū (神刀流)
  are collectively called Kashima seven styles (鹿島七流).”　

(Reference: http://jigenryu.jp/shobukan/newpage15.html)