Kashima Shinden Jiki Shinkage-ryu Kenjutsu (鹿島神傳直心影流 剣術) is one of the most powerful forms of Japanese swordsmanship. It places emphasis on posture, breathing, focusing the mind and spirit (kiai), and developing a powerful cutting ability. It has a very austere character, with kata that are very strenuous to perform and develop a hard form of internal power. Jiki Shinkage-ryu regards itself as the "true" Shinkage-ryu. Its fourth headmaster, Ogasawara Genshinsai Minamoto no Nagaharu (小笠原源信斎源長冶, 1574–1644), spent 20 years in Beijing in the early 17th century, practicing Chinese martial arts, including the large and heavy kwan dao. Its 14th headmaster, Sakakibara Kenkichi (榊原鍵吉, 1830–1894), was bodyguard to the Shogun and keeper of Edo castle. Jiki Shinkage-ryu contains a profound training regimen focused around the development of kiai (気合), which can be regarded as the active complement of aiki (合気).
In Jiki Shinkage-ryu, the foundational practices of suburi (cutting drills) and unpo (walking practice) allow practitioners to begin to learn the foundational practice of Hōjō no Kata (法定之形) or "Four Seasons" kata. Hōjō is an extremely intense practice that develops posture, distance, timing, spirit, and power in a swordsman, and provides the proper foundation for learning the strategy and tactics of the system. The practice of Hojo was said by Yamaoka Tesshu to be a form of zazen. Another famous practitioner of Jiki and student of Sakakibara was Sokaku Takeda, reviver of Daitō-ryu Aiki-jujutsu and teacher of the founder of Aikidō, Ueshiba Morihei.
Aiki Inyo Ho (合氣陰陽法) is a named taken from a neo-Confucian school, and refers to this development of the body and spirit, leveraging Taoist principles. I believe it is an apt summary of the practices of Jiki Shinkage-ryu, as Jiki Shinkage-ryu incorporates five element theory and the theory of yin and yang in its practice. In that regard, I find Jiki Shinkage-ryu to be a form of internal training that moves from hard to soft, similar in some ways to Xingyi Quan.
Once the body and spirit are developed sufficiently, the strategy and tactics of the art are taught as part of To No Kata, typically performed with fukuro shinai to allow for full power practice within combative range. These kata are practiced with spring kiai. Gekken can be introduced at this time, as well as henka waza that explore each kata fully. At higher levels of practice, a very aggressive set of kata with the small sword (kodachi) are taught, using summer kiai. Timing, distance, power, and balance are stressed, and the notion of kuzushi or off-balancing. Eventually, students are introduced to formalized paired practice with metal swords (habiki), using autumn kiai, that further develops internal principles in the practitioner and introduces key strategies of the art.
The distinction between the Jiki lines of Shinkage-ryu and the Yagyu lines is clear, but they are likely driven by the same essence. The word "jiki" means "correct" or "true". The emphasis on kiai and dominant spirit in Jiki are an outward (omote) manifestation of some of the essences of Shinkage-ryu. Because Shinkage-ryu grows out of a deep study of and critique of Shinto-ryu by its founder (Kamiizumi Ise No Kami), it is said to contain the gokui or essence of Shinto-ryu in its foundational kata (empi no tachi). This kata is a core part of Yagyu Shinkage-ryu -- it was the first set taught by Kamiizumi to his students, and other kata were developed later to explain and elaborate on its principles. Empi is curiously absent in Jiki's formalism, but key elements of it can be found distributed throughout Jiki Shinkage-ryu, if you look hard enough.
The founder of Jiki Shinkage-ryu, Matsumoto Bizen-no-kami was one of the earliest students of Kamiizumi Ise-no-kami, and Jiki Shinkage-ryu does not contain the foundational kata of empi that is crucial to other lines of the art. It may be that Matsumoto's study with Kamiizumi predates the development of that set of teachings.
At the same time, Jiki Shinkage-ryu, as it survives today, has far fewer techniques than other systems of kenjutsu. A valid question is whether it has been pared down too much over time. Up until the late 19th century, however, Jiki Shinkage-ryu was a sogo-bujutsu, and included in addition to tachi and kodachi the practice of bo, naginata, yari, and yawara (grappling) in its curriculum. A focus on shinai sparring by its 14th headmaster, Sakakibara, led to eventual loss of those additional practices. However, it is interesting to note that once experience is gained in Hojo and To no Kata, those same forms can be practiced with odachi, bo, and naginata without great modification. Additionally, the kodachi kata teach stability and kuzushi from tsuba zerai, and introduces ideas related to work at close quarters with edged weapons to the practitioner. Because of the prior emphasis on shiai, and the condensing of Jiki Shinkage-ryu's curriculum, it is of great benefit to explore applications of To no Kata in free practice. By the late 20th century, the main line of the art did not practice shiai openly, so we as contemporary practitioners are left to explore applications of the kata largely based on direct experience in both kata and free practice and intuition driven from the practice or study of related arts.
What's in a name?
Not much, or a great deal, based on your perspective.
Glossing Jiki, Shin, and Kage as a single word in English seems to give it the character of a proper noun "jikishinkage" as the name of the school. Splitting Jiki from Shinkage, as done in the group I train with, puts the emphasis on Jiki meaning "true" or "correct". We are doing the correct Shinkage-ryu, as opposed to that other Shinkage-ryu over there. In fact, historically, this is the rationale for the choice of name.
The character we use for shin is read as heart, from the Buddhist concept of Jikishin, or original/pure heart, the bodhi seed of enlightenment. This, as I mentioned, is the focus placed by the practitioners of naginata, but it bears rememberance. Yamaoka Tesshu, the famous swordsman, calligrapher, and Zen priest of the late 19th century is known to have said that if one practices Hojo, the foundational kata of Jiki Shinkage-ryu, it is the equivalent of practicing zazen.
It is strange how names work on the mind. Despite agreeing with that sentiment, I have heard someone who practiced the naginata style using the same name (popularized by Sakakibara's wife), refer to the art as "jikishin" for short, putting an emphasis on that original mind of Zen. At first I didn't know what he was talking about - I had to do a double take, even though I knew the story of Tesshu's comments well. We had always referred to our art as "Jiki" in short. In our minds, we practice the "true" Shinkage-ryu.
Kage is glossed as "shadow" and at times the character for yin has been used in its place. There is a notion of matching, moving with, and also solidity to the art when practiced in depth. Also, the notion of invisibility or concealment, either of your person or your intensions, and the protection that affords (specifically, as granted by the patron warrior deity Marishiten) comes into play when speaking about the concept of shadow in Japanese swordsmanship.
Often I will just say I practice Shinkage-ryu, inspired by the story of a budo gathering after a large enbu (demonstration), when a senior from my teacher's dojo introduced himself simply as "Shinkage-ryu Kato-desu": I am Kato, of Shinkage-ryu.
The true art does not need to differentiate itself from others, it is simply Shinkage-ryu.