The invention provides seed and plants of the spinach variety designated CALLISTO. The invention thus relates to the plants, seeds and tissue cultures of spinach variety CALLISTO, and to methods for producing a spinach plant produced by crossing a plant of spinach variety CALLISTO with itself or with another spinach plant, such as a plant of another variety. The invention further relates to seeds and plants produced by such crossing. The invention further relates to parts of a plant of spinach variety CALLISTO, including the leaves of such plants.