Prudence carries two objects in Western art and they say two different things about her. The mirror is self knowledge: the virtue of practical judgement has to begin with an accurate view of the person doing the judging, and a mirror is the shortest way to draw that. The serpent comes from Matthew 10:16, where the disciples are told to be wise as serpents and harmless as doves, and it supplies the quality the mirror does not, which is wariness. Together they make prudence the only virtue whose emblems include an animal usually reserved for the vices.
She often has a third attribute, and it is the most interesting one. Late medieval and Renaissance painters gave prudence two or three faces, looking in different directions, because the tradition defines her by the three parts of time: memory of what has happened, attention to what is happening, and foresight of what may. Titian's Allegory of Prudence is built entirely on that scheme, three heads above three animals, with an inscription about learning from the past so as not to spoil the future.
The three faced form is also why prudence occasionally unsettles viewers who meet her without the key. A woman with a snake and several faces reads as a monster until the programme is explained.
The atlas draws the mirror with the serpent, which is the pairing that stayed standard, and leaves the multiple faces to the paintings that can carry them.