Mutable is the modern name. The older English one is common, translating the Latin term for what is shared, and it means a sign belonging to two seasons rather than to one. Double bodied is the direct rendering of the Greek. All three are in print and all three name the same group.
The four stand at the ends of a cross, each in square or opposition to the other three, in the same way the cardinal and fixed groups do. Gemini and Sagittarius form the axis of learning near and far, and Virgo and Pisces the axis of service and dissolution, which is a modern formulation of something the older house lists imply without stating it.
There is a further technical use worth recording. In horary practice, a mutable sign on the relevant cusp is read as a sign of repetition or return: a matter that will recur, a person who will come back, a thing not finally settled. A cardinal sign on the same cusp is read for a quick outcome and a fixed sign for a lasting or immovable one. That triad of judgements is one of the places where the qualities do concrete work rather than describing temperament, and it belongs to the horary literature rather than the natal.
Unlike the cardinal ingresses, the Sun's entry into a mutable sign has no astronomical event attached to it. The group is defined by its position within the season, and that definition is the tradition's own rather than the sky's.
Sources Christian AstrologyWilliam Lilly with TetrabiblosClaudius Ptolemy