As Shinigami come from Heaven, so do Onimorte come from hell.
Most of the time, these people stay within the realm of hell, making sure that appropriate torments and punishments are delivered to the sinners, as well as making sure that they are rehabilitated. Onimorte are many things, but evil they are definately not. If they seem to be taking too much pleasure from delivering the punishements they deliver, they may well end up receving the same from their fellows. In general, they are a very carefull bunch, since it takes less transgressions for them than others to end up in the Pit.
Of course, breakouts, while rare, are not unknown. This is mostly when the agents of hell are unleashed en masse upon the world. They are assigned to hunt these fell souls as they run. Sometimes, if the soul in question is nearly pure anyway, they will simply let it go, and allow the Shinigamui to shepherd them on to Soul Society. However, if any of these escapees, no matter what their status, become hollows, it's game on: They will be hunted. These occasions are the times when Shinigami and Onimorte usually meet. Of course, these ones are usually quite smart (for hollows) and desperate, so Shinigami sometimes steal the strike. But the gates of hell will always close upon them.
Onimorte use Limbus Arts to help with their duties - usually by destabilising parts of a target's nerve-systems, causing some incredible ammount of pain. Other torments are caused by altering the elerctrical flows within the brain, creating some horrifying hallucinations, which can seem almost completely real. A few halucinations are banned, such as making the sinner beleive they have acheived salvation before their time and simulating their release, only have reality crash back with force. there are at least two Onimorte still serving pennance for this crime.
Onimorte use weapopns called voidblades - these have small gateways into oblivion trapped within them, and it is the force of non-existance which causes alot of the mysical damage these weapons inflict. The binding is kept going by a sentient force, similar to that of a Zanpakouto. It has a great deal of power, and is kept heavily in check. This demon (as it is often refered to) will lend it's power to one who knows it's name - of course, they are often reticent about sharing this information. Once the name of the "demon" is known, it can be bound to the weilder, granting powers comparable to shikai. When both the weilder and weapon are strong enough, they can attempt to weild the power of the void, with the technique called Siburi Kiai. Even when they know it, it's use is discouraged, since they are potentially increasing the force of entropy in the world.

Onimorte are set out in six divisions, four of which correspond to the classical elements:

Ao- The Ao division (Standing for All One) is the governing force of the Onimorte. All administrative duties are carried out by the lower ranks of this division. Trainee onimorte start in the Ao divison, and then proceed to the others to finish. The central comand of the Onimorte is a 25-strong comitee, often called "The Light Of Heaven", although it's actual name is some long-winded beaureucratic title that is often forgotten.
Ogre- The Ogre division is styled after the element of earth. This division is never fast, but when a decision is made, it will stick to it, and carry it through. Often nicknamed as "unmovables".
Tengu - The Tengu division is styled after the element of air. Regarded as the most hasty division, this division also encompasses the Onimorte inteligence gathering services, and has many high-speed scout teams.
Shark - The Shark division is styled after the element of water. The more medically inclided division, this lot also have a research & development institute, although it is not as prolific as the shinigami equivilent. They are called sharks, because they are still able to do a great deal of damage when the moment is called for.
Kumo - The Kumo division is styled after the element of fire. They are the guardians of the Pit where the sinners are kept, and it is their duty to assist in their charge's rehabilitation. these spiders don't like people escaping their webs.
Kitsune - This division is full of all those who don't really fit in with the others. Tricksters, loose cannons, mavericks, anyone with slightly unorthodox methods (but methods that are not flat out illegal) can usually be found here. The Foxes are called upon to do the job when protocol and procedure fail. There is no formal command structure in the Kitune division, but one based on relevant experience and power.

Nomenclature: French inspired
Garb: White robes/ceremonial garments, any colour trimming
Onimorte also occasionally grow horns. y'know. Demony-style.