I know how to recognize males and females (based on flowers), but how can I recognize a hermaphrodite papaya?
The hermaphrodite papaya is recognized by its flowers. They have both male and female characteristics, which is why they're able to self-pollinate. Unfortunately, that means you really can't tell the gender of the tree until between 3 to 6 months after transplanting, when it begins flowering.
From the College of Tropical Agriculture at The University of Hawaii:
Hermaphrodite flowers have both an ovary and stamens bearing pollen. They can pollinate themselves and do not require the presence nearby of another papaya plant. They are borne in the leaf axils, like the female papaya flowers.
According to Hawai'i Homegrown Food Network:
Hermaphrodite papaya trees have fruits with seeds. The flowers are thinner than females, with a tubular shape. Hermaphrodite papaya flowers contain both male and female parts within each flower, very close together. This arrangement nearly always ensures that the flowers will self-pollinate. The male part (stamen) has orange pollen-containing anthers on the ends of short white filaments. The pollen can drip right down onto the female part of the flower (pistil). Thus, all hermaphrodites bear fruit. The fruit is elongated, somewhat football-shaped, compared to the rounder fruit the female trees produce.
Below is the female papaya flower. Note the long petals and fat ovary with branching yellow stigma on top. Credit: Richard Manshardt 2014.
Below is the hermaphrodite papaya flower. Note the orange male anthers, and cylindrical ovary with no branching on top. Credit: Richard Manshardt 2014. Source.
Another good scientific resource is the Food and Fertilizer Technology Center in Taiwan.
From Genetics and Genomics of Papaya, Chapter 3:
(c) Male (d) Male fruit-bearing plant