The gyaku-en indicates the concept of not believing in the teachings of Buddha (i.e., defying the 'en', or connection with Buddha) unquestioningly or someone who can not be saved. It is the antonym of the word jun-en. The gyaku-en also refers to the act of criticizing Buddhism first and yet being led to become a monk all the more for the criticism.
The gyaku-en originally meant the act of opposing Buddhism by defying fate and order as well as by making unethical remarks and committing unethical acts; subsequently, however, the word also began to indicate cases where criticizing Buddhism triggered inversely one to become a monk. In some cases, the gyaku-en refers to the fate that interferes with one's ascetic practices.
The incident 'Gosenkiko, or Gosenkikyo' in which 5,000 arrogant ascetic monks, believing to have become enlightened, left before being taught the crucial lessons in the chapter two of the Buddha's words in the Hoke-kyo Sutra (the Lotus Sutra), is said to have been a classic example of the gyaku-en.
Additionally, there is a famous allegorical story called Dokku no En in the Nehan-gyo Sutra (the Sutra of the Great Nirvana). The legend holds that the dokku was a drum to which poison was applied, and that when the drum was beaten, all the people hearing the sound would die; in the same way, the teachings of Buddha are to eliminate the three poisons of avarice, anger and ignorance among people.
The Kamidaigo-ji Temple in Kyoto is counted as the eleventh in the Saigoku Kannon Reijo, or the holy places of Kannon (Goddess of Mercy) in the Western Provinces, and its goeika (song in praise of the Buddha) reads, 'If there should be a prayer to save even all the gyaku-en, how dependable the Junteido (main sanctuary of the Kamidaigo-ji Temple) would be!"
The gyaku-en also referred to cases in which one's archenemy or unrelated person held a memorial service for him/her or the older people held a memorial service for the younger ones; consequently, the meaning of the gyaku-en was converted to commonly refer to the case where a child died before his/her parents.