Quadrants

It seems the lack of reproductive sex just makes relationships more complicated. Troll culture is built around a set of extremely complicated relationships. As Karkat says to Vriska, trolls are driven by the two primary emotions, hate and pity. (Keep in mind it's Karkat of all people that's saying this.) It seems like both of these are necessary in troll society, although most of their social system remains obscure.

Known types of relationships include mating fondness, moirallegiance, kismesissitude, and auspisticism. Nepeta appears to be keeping track of this with her shipping wall. If this is the case, hearts would likely denote matespritship and diamonds would possibly cover moirallegiances.

Red Romance (Redrom)
These types fall under the Pity hemisphere of Troll emotions, and are associated with the color red.

Mating Fondness/Matespritship ♥
While trolls do not breed in the same fashion as most earth creatures, they nevertheless seem to recognize attraction or possibly even love under the heading "mating fondness". Whether trolls actually experience mating fondness or simply recognize it in the lesser creatures (like humans) is a bit unclear, but Karkat has used the term as a point of comparison. Mating fondness corresponds with the ancient Greek concept of έρως (érōs ).

Speculation:
 * Equius and Aradia; this may or may not be two-way.
 * Nepeta for Karkat, according to her shipping wall.
 * Vriska for Tavros, mostly because if CT and AA count, then its about the same emotion.
 * caligulasAquarium for Terezi
 * Sollux and Kanaya, AH said that this was likely on formspring.
 * cuttlefishCuller for Gamzee

Moirallegiance ♦
GA: It Just Means Theres Got To Be Someone Around To Keep An Eye On Them

Moirallegiance (derived from the three Moirae, or Fates, and the word allegiance) is a process that occurs between a troll and their "moirail". Moirallegiance appears to be a sort of fated friendship or guardianship. Moirallegiance is linked to pity in the same way that black romance is linked to hate. 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 as a type of romance.

Moirallegiance may be analogous to the ancient Greek concept of αγάπη (agápē ).

Known moirails:
 * CuttlefishCuller's is CaligulasAquarium
 * Kanaya's is Vriska
 * Nepeta's is Equius

Black Romance (Blackrom)
These types fall under the Hate hemisphere, and are associated with the color black.

Auspistice ♣
CA: i figure if youre going to auspisticize any twwo brinesuckers wwho sneer at each other a funny wway you might as wwell make it official and be ours right

An auspistice (derived from the terms auspice and armistice) is a "facilitator" of some sort between two others, mediating interactions between them. CA said that he and Vriska would likely become Kismesisses if Kanaya did not intervene and Auspistice for them, which hints at the relationship between the two mediated parties being similar-- but distinct-- from a Kismesis relationship.


 * Kanaya between CaligulasAquarium and Vriska. According to CA, Kanaya can be called upon frequently for this function, and he refers to her as "the village twwo wwheel devvice when it comes to auspisticing."
 * John almost gets caught in this between Terezi and Karkat.

Kismesissitude ♠
This is what a troll feels towards someone known as their kismesis (derived from the terms kismet + nemesis) or "fated enemy". This is linked to hatred, and it is necessary that both parties feel hatred towards each other. It is unconfirmed whether any of the revealed romances between troll characters fall under this category, although Aradia and Equius are likely candidates considering she beats the shit out of him and then kisses him, and he finds her red blood disgusting yet somehow tantalizing. Karkat speculated that Vriska had (unrequited) Kismesis-like feelings toward Tavros.

Kismesis may be analogous to the ancient Greek concept of στοργή (storgē ).

Friendship
A troll disease which spreads among groups if not carefully controlled. It should be noted that in troll language, the word for friend is exactly the same as the word for enemy. (or, at least that's what Equius says, even though he says both words separately, in the same conversation, no less). It is possible that friendship may overlap with the four romances.
 * Karkat is Gamzee's best friend and hates every minute of it.
 * Sollux and Karkat are friends as well. For such a crab, Karkat has a lot of friends.

Friendship corresponds with the ancient Greek concept of φιλία (philía).