Brief description of Shinkage-ryu kata and their relationship
Sangakuen no tachi and Hojo originally were versions of the same practice, but the corresponding kata have diverged quite a bit over four hundred years. Matsumoto Bizen no Kami, the founder of Jiki Shinkage-ryu, was an early student of Kamiizumi Ise no Kami Nobutsuna, and the version of Sangakuen he learned may have differed from other approaches, or the kata were changed substantially by the fourth headmaster, Ogasawara Genshinsai Minamoto no Nagaharu (小笠原源 信斎源 長冶, 1574–1644), who obtained influences on his practice from his long time in China.
Empi no tachi is believed to contain counters to the essence (gokui) of Shinto-ryu. It is as a result helpful, if possible, to practice or observe and analyze versions of Shinto-ryu kata sets such as Gogyo-no-tachi or Shichijo-no-tachi to better understand what Empi-no-tachi might be concerned with. Because Kamiizumi Ise No Kami first taught experienced swordsmen he had defeated in combat, but who showed promise, empi no tachi can be hard to appreciate if one is not already skilled with the sword. So, sangakuen-no-tachi and nanatachi are sometimes said to be practices developed to help explain or elaborate on empi.
Sangakuen-no-tachi is also said to counter the introductory set of Shinto-ryu teachings called Itsutsu-no-tachi. Likewise, Nanatachi is said to be focused on countering the next set of Shinto-ryu teachings called Nanatsu-no-tachi. However, this can be hard to explicitly see in how both Shinkage-ryu and Shinto-ryu are practiced today, as they have diverged quite a bit over the last several hundred years. In both cases, these explanatory kata are said to develop skill at defense in the Shinkage-ryu practitioner.
Kuka no tachi is said to be instead about learning how to attack, and is said to contain methods of defeating the upper-level teachings of three popular or important ryuha from the 16th century: Shinto-ryu, Nen-ryu, and Chujo-ryu (which eventually developed into Itto-ryu). However, it is not easy to see what kata the uchi role corresponds to which of the three ryuha listed above.
Tengusho is an advanced practice that supposedly was also practiced by certain lines of Jiki Shinkage-ryu until the late 19th century, so we maintain a version of it in our practice. Tengusho deals with bringing a new order from chaos and brings Shinkage-ryu practice to a more immediate, intuitive, or divine level of practice.
Shugendo: Taosim, Buddhism, and Shamanism
Shugendo is a name for a uniquely Japanese mixture of Taoism, Esoteric Buddhism (Mikkyo), and shamanistic practices akin to Shinto and other mountain religions, which influenced Japanese swordsmanship greatly until the Meiji era.
Shinkage-ryu kata can all be related to philosophical concepts, beyond the typical association with Zen Buddhism. The concepts of the three treasures (sangen or sancai) of Heaven (Ten), Earth (Chi), Humanity (Jin) and Spirit (Shin/Shen), Mind or Intention (I/Yi), and Intrinsic Energy (Ki/Qi), as well as the principles of complementarity (Inyo/Yin-Yang) and the the five Taoist elements (Gogyo/Wuxing) all have meaning in a Shinkage-ryu context. These mappings happen when we examine individual kata in each named set, as well as the relationship of the kata sets to one another.
I believe these mappings are something Jiki Shinkage-ryu and other lines of Shinkage-ryu have in common, but the mapping is more explicit and clear in Jiki Shinkage-ryu than other arts. It is for these reasons I feel that examining internal martial arts from China, which make many of these concepts explicit and primary in their practice, versus hidden or obscured, is so beneficial to the practice of Shinkage-ryu.