Final answer:
The form of government in Qing China that included both native Chinese and Manchu invaders was the imperial bureaucratic system influenced by Confucian principles, featuring dual-administration in major ministries.
Step-by-step explanation:
The form of government in Qing China that offered a position in the local government to the native Chinese, as well as the Manchu invaders, was the imperial bureaucratic system established especially during the reign of the first Qing emperor, Kangxi. This arrangement was derived from the Confucian principles of governance and maintained through the system of imperial examinations. Both Manchu and Han Chinese co-administrators led each of the government's six major ministries, signifying a dual-administration system which promoted a degree of unity and cooperation between the Manchus and the Chinese.
The Qing Dynastic Cycle does not refer to a specific form of government, but rather to the period during which the Qing Dynasty ruled China. The term 'Confucius System' can misleadingly imply a formal system of governance directly attributed to Confucius, which is not the case. Communism and fascism are unrelated to the Qing governance, as they are political ideologies that emerged much later than the Qing period.