Gemini, Virgo, Sagittarius and Pisces are the signs the Sun occupies in the last month of each season, when one season is giving way to the next. Ptolemy calls them bicorporeal, disoma, meaning two bodied, and the reason he gives is that each shares the nature of the season it ends and the season it begins.
That is a stronger claim than the modern word mutable conveys. Bicorporeal says these signs have two natures at once, not that they change easily, and the tradition applies the term literally: three of the four are represented by double figures, the twins of Gemini, the two fish of Pisces, and the horse and man of Sagittarius. Virgo is the exception, and the reason usually given for including it is its position in the season rather than its image.
The reading that follows is adaptation, distribution and preparation. A mutable sign does not begin a thing and does not hold it; it carries it across, dividing and passing on what the fixed sign accumulated. The tradition also gives the group flexibility and inconstancy as two readings of one property, and does not treat either as the correct one.
Mercury and Jupiter rule the four between them in the traditional scheme, Mercury taking Gemini and Virgo and Jupiter taking Sagittarius and Pisces. It is a neat symmetry and the sources notice it.