Your satsumas are probably grafted to trifolia rootstock. This is done to allow the plant to go into semi-dormancy and survive cold snaps (although it is less necessary with satsumas). Trim the long thorny branches back towards the trunk to encourage the growth of the "satsuma" branches. Never let the thorny branches grow out of control or you may only have a trifolia (bitter orange).

Cut the thorny part of flush with the trunk of the satsuma tree. These are understock and will become the main part of the tree if allowed to grow.

Hello I am also new to the group. I live in Louisiana and have the same thorny growth on my new trees (planted this spring). I see someone has suggested to cut them off, however, on my trees, the new thorny growth is coming right off the ends of the existing growth and not off the trunk. Am I still supposed to remove these thorny growths?
Thankx
Bill