Jabberwocky and Bandersnatch will regularly heal each other to full health. Jabberwocky is immune to magic and takes half damage from physical attacks, but can be inflicted with Deprotect, Slow and Poison. Jabberwocky is weak against all elements.
The best strategy is to pick a target and kill one of them quickly. A Synergist who knows En- spells can cast them on the Commandos to deal double the regular damage. Just because a party member isn't dealing damage to a target doesn't mean they are not building its chain gauge. Fang's Full ATB skill can wipe the Jabberwocky out quickly if it has a decent chain gauge, especially if Fang is buffed with Bravery, or better, Bravera.
Jabberwocky is a nonsense poem written by Lewis Carroll in his 1871 novel Through the Looking-Glass, and What Alice Found There, a sequel to Alice's Adventures in Wonderland. In an early scene in which she first encounters the chess piece characters White King and White Queen, Alice finds a book written in a seemingly unintelligible language. Realising she is travelling through an inverted world, she recognises the verse are written in mirror-writing and holds a mirror to one of the poems, and reads the reflected verse of "Jabberwocky." She finds the nonsense verse as puzzling as the odd land she has passed into, later revealed as a dreamscape. The creature called Jabberwock is never fully described but is accepted to be an unworldly and vicious entity.
Bandersnatch it fights alongside with also relates to this poem.