The Kuinshī are the antithesis to the Todgott, Shinigami who have obtained Quincy powers. Having begun as Shinigami, the Kuinshī appear entirely human, and barring those times where they are using their powers are indistinguishable from ordinary Shinigami. To gain access to their Quincy powers, a Kuinshī dons a Quincy cross. This allows them to augment their Shinigami abilities with Kuinshī powers, as well as giving them access to some of the generic Quincy abilities. Their spiritual pressure likewise changes into a mix of Quincy and Shinigami.
It is unclear what the Kuinshī are after. The current member of the Kuinshī has been shown to be a former Lieutenant class Shinigami. They are considered an oddity, but not an enemy, by the Soul Society. The Kuinshī are rejected by Quincies, due to their former status as Shinigami. It is because of this that it can be presumed that the only side the Kuinshī are on is Soul Society's. Regardless, they seem to be valuable allies-to-be in the battle against Hollows and Arrancars, as Ryan Getsueikirite mentioned. The Kuinshī have not shown any sort of hierarchy, seeming to prefer a loose-knit organization.
Powers and Abilities
Quincy Cross (滅却十字, mekkyaku jūji): The Kuinshī remain in complete control while wearing using their crosses, the crosses are a requirement to use their Quincy Bows. As suggested by Matthew Toshiro, the fusion of the two spiritual entities allows these Shinigami to reach levels of power far surpassing that of either Quincy or Shinigami.
The Kuinshī use their Quincy powers to augment their existing Shinigami abilities, granting them massive increases in speed, stamina, and strength. Also their bow's focal point is the Quincy cross, acting as the focus for a Quincy bow. The cross need not necessarily be in a cross shape. Matt's first Quincy cross took the shape of a cross, but his recent one also takes the shape of a pentacle.
Quincy Bow: Every bow has an appearance unique to it's owner, but seemingly only once they've developed more power. Standard Kuinshī archers from the past were seen using bows almost identical to Uryū's first bow, as well as his grandfather's. Each bow is a manifestation of the Kuinshī's power.
Reiryoku Absorption: The Kuinshī fuel their abilities by absorbing the Reiryoku of their surroundings in the form of spirit particles. They primarily absorb them from the atmosphere, and have an easier time collecting this energy in environments with higher concentrations of spiritual particles, such as Soul Society or Hueco Mundo. To a sufficiently powerful Kuinshī, there is no difference between spiritual particles and spiritual energy as a source of power.
Reiryoku Manipulation: one can manipulate spiritual energy for offensive purposes. They most commonly reshapes spiritual particles into various types of bow and arrows, but can also concentrate spiritual energy into the form of swords or explosives.
Hirenkyaku: Analogous to the Shinigami Flash Steps or the Arrancar Sonído, the Kuinshī Hirenkyaku allows the user to move quickly over short distances; according to Uryū Ishida it is faster than flash steps, though Mayuri Kurotsuchi's superior speed suggests that this is just a generalization. To perform the technique, the user gathers spiritual particles under their feet and rides them to the desired location. Like the air-walking skill displayed by most Arrancar and Shinigami, it can also be used to hover or fly for long distances.
Matthew gained his current Kuinshī status due to his interaction with a Sanrei Glove during his bankai training. It is unclear how other Kuinshī gain their Quincy powers. A possible second process used seems to spread a some form of possible airborne infection that has the desired effects on high-level powered shinigami as compared to low-level powered ones. Causing those infected great increases of reiatsu absorption, until it leaks out and forms a Quincy cross.