Greek fate language is craft language before it is theology. A moira is a portion, the part that falls to somebody when a thing is divided, and it is used of a share of meat at a feast long before it is used of a lifespan. Clotho is the point at which that portion is manufactured. Her name comes from the ordinary verb for spinning, and spinning in Greek life is domestic, continuous, and done by women: fibre is drawn from a distaff, twisted onto a spindle, and wound. The metaphor is old enough that Homer can speak of what was spun for a man on the day his mother bore him without needing to explain it, which suggests the image came first and the three goddesses were fitted to it afterwards.
What the image carries is a specific idea about how a life is limited. A thread is not cut short by accident; it is made to a length, and the length is settled while it is being made. That is why Clotho belongs at the beginning of the sequence rather than at the end, and why she is the one invoked at births and, in some places, at marriages. Both are moments at which an allotment is being fixed rather than run out.
In art she is grave rather than beautiful, usually shown with distaff or spindle, standing with her two sisters. Rome absorbed the group as the Parcae and gave them counting names, Nona, Decima and Morta, the ninth month, the tenth month and death, which is the same structure rendered as a calendar of gestation rather than a piece of thread.