Matespritship
These types fall under the pity hemisphere of troll emotions, and are associated with the color red. They are tied to strongly positive emotions.
The one most like human love/romance. Matesprits have a strong feeling of affection for one another - though troll psychology being what it is, courtship can be just as violent as that between kismeses, and indeed it can flail between one and the other. It is one of the two concupiscent relationships, those that deal with reproduction, the other being kismesissitude. Matespritships's quadrant is known as the flushed quadrant.
Moirallegiance/Pale Quadrant
A troll is compelled by fate to watch over their moirail and keep them in line. Despite the platonic appearance of this role (as we currently understand it) to us humans, trolls consider it a type of romance. Moirallegiance may also entail pacifying one's partner, like what Nepeta does for Equius and Karkat for Gamzee. Moirallegiance may translate (roughly) to our human concept of "love for siblings", which contains no intent for reproduction.
Auspisticism/Ashen Quadrant
An auspistice is a "facilitator" of some sort between two others, mediating interactions between them and keeping their relationship functional. Auspisticized pairs include Eridan and Vriska, Spades and Snowman, and Vriska and Tavros, all of whom have been shown to be potentially caliginous to some degree, suggesting the emotions between two auspisticized trolls are similar to but distinct from those of kismeses.
Kismesissitude/Caliginous Quadrant
This is what a troll feels towards someone known as their kismesis. This is linked to hatred, and it is necessary that both parties feel hatred towards each other. Not any kind of hatred will do, since there is such a thing as platonic hatred amongst trolls (similar to the human concept of platonic love).In other words, kismesissitude seems to be based on a mix of hatred and sexual attraction. As it thrives on a rivalry between two trolls, defeating or killing your kismesis is discouraged, since obviously there wouldn't be a relationship anymore