The 'both-ways-liberated one', is the name of one class of Noble Disciples ariya-puggala. He is liberated in 2 ways, namely, by way of all 8 absorptions jhāna as well as by the supra-mundane path sotāpatti, etc. based on insight vipassanā. In M. 70 it is said:
Who, o Bhikkhus, is a both-ways-liberated one'? If someone in his own person has reached the 8 liberations absorptions, and through wise penetration the fermentations āsava have become extinguished, such a one is called a both-ways-liberated one. Cf. D. 15.
In the widest sense, one is both-ways-liberated if one has reached one or the other of the absorptions, and one or the other of the supra-mundane paths cf. A. IX, 44.
The first liberation is also called 'liberation of mind' ceto-vimutti the latter liberation through understanding' paññā-vimutti.
The first liberation, however, is merely temporary, being a liberation through repression vikkhambhana-vimutti = vikkhambhana-pahāna