Nestorians are a sect of followers of Nestorius who deny the hypostatic union and maintain the existence of two persons in Christ; the man Jesus and the divine Son of God, rather than as one divine person, unified, with two natures. Hence Nestorianism has been considered heretical by most Western and Orthodox churches.
Nestorius (c. 386–451) was Archbishop of Constantinople and his view of Christ led to the Nestorian schism, separating the Assyrian Church of the East from the Byzantine Church.
The difference twixt Christ as two persons or one person, is most significant for the Christian's understanding of salvation and the theology of the Eucharist. Nestorians rejected the concept that it was God who was crucified; rather it was the humanity of Christ who suffered.
Nestorians were active missionaries and in the 7th and 8th centuries spread Christianity eastwards to Persia, Central Asia and East Asia, including China, Mongolia, Korea and Japan. Their influence waned and the Church was crushed in the Far Eastern realms but remained relatively strong in the rest of Asia.
As for the above so-called Nestorian Cross, variations of this budded style have been found on Nestorian or Assyrian tombs in Asia.
However, it should be noted that centuries ago, in Nestorianism's heyday, it's unlikely that there was a single 'corporate logo' for the religion. The image above is a cross design influenced by the art of local culture and tradition.
Click the image on the right to see a cross discovered in Mongolia in the early 20th century, it includes a swastika in the centre; a symbol used in Buddhism. This central positioning does not suggest to us integration of Buddhist and Christian religions; rather the swastika is seen as a spiritual symbol.
You will also notice the four birds, whose legs make eight divisions in the symbol. The number eight is associated with spiritual rebirth.
The original significance of the four birds is unknown. They could relate to a story similar to that of 3rd century St. Vincent, or to any part of the Bible that refers to birds. We feel it's unlikely they represent the Holy Spirit, who is usually represented by a single dove, as is the Genesis story of Noah. Alternatively, like the swastika, the birds could be symbols from an early Asian religion, such as Tengrism or some other shamanistic cult where birds are considered to be spirit messengers.
As with all symbols, whatever the original significance of this cross was, it is of lesser importance than what it means to us personally.