Taiji used to be Romanised as "T'ai chi", though the martial "T'ai Chi" familiar in Europe in only one small part of it.
Taiji is torn between li and wu wei.
Li is tradition, etiquette and proper ceremony in accord with the ways of the ancestors. It characterises Ruism ("Confucianism") which is scholarly (ru denotes scholarship) and politically active.
Wu wei, literally "without action", denotes relaxation and spontaneity. It characterises Daoism ("Taoism") which is meditative and more politically passive.
Ruism also speaks much of dao, which was an ordinary word for "path", but "Daoism" specifically denotes thinking akin to the Daodejing.
Both li and wu wei affirm the complementarity of yin and yang.
The saying "Ruist outside, Daoist inside" claims the two elementary attitudes as complements, but really they are incompatible.
Taiji dominated Chinese culture for two thousand years, and remains influential despite Humanist incursions in the 20th century CE.
Since about 100 CE Taiji has faced competition in China from the Buddhist form of Yoga, imported from India, but as far as I know Taiji and Yoga have never been compounded together; Neoconfucianism (from about 1000 CE) was a reinvigorated Taiji, assimilating some Buddhist ideas without, I think, compromising on essentials.
Back to The Major Cogniframes.