It depends from which part of the tree the budwood was harvested. I've originally got my Tarocco budwoods that have very small thorns, and so their branches have almost no thorns on my tree. It is now occupying about 15% of the canopy volume of my multi-grafted pigmented orange tree and haven't seen any thorn that are big, there are few, less than 1/8" long but mostly none at all. It is one of the best tasting blood oranges that I have.
I also have what appears to be a mutation of the Tarocco blood orange. On one branch, the fruits consistently have Navels on them, and have propagated that to other friends, next year I would have some results.
Also, take note that there are several Tarocco blood oranges available from UCR CCPP. The best ones that they told me is the Tarocco from Bream. The problem is that the only foundation stock tree is not inside the screenhouse, and it is quarantined due to outbreak of Tristeza. So I wasn't able to obtain the Bream Tarocco. Perhaps the Bream Tarocco has thorns, but it is said to be tastier even in cooler climates like the coastal areas.