The Julian Society is (or was) a religious order dedicated to the memory of Julian the Apostate, last pagan emperor of the Roman Empire. Why do I say “was”? Because the last new entry on the Society's blog is from 2009, so the extent to which they are still an active organization is unclear.
Although the emperor Julian was indeed a pagan, his paganism was distinctly different from that of the ancient Greco-Roman world. Julian was steeped in the doctrines of Neo-Platonism, a tradition of mystical philosophy that had both a pagan faction (represented by philosophers such as Plotinus, Porphyry and Iamblichus) and a Christian faction (represented by philosophers such as Origen and Pseudo-Dionysius). Christian Neo-Platonism had a huge influence on the development of Christian doctrine, and some of the tenets of pagan Neo-Platonism were a lot more similar to Christianity than most neo-pagans would be comfortable with. For instance, Neo-Platonists of both factions tended to advocate an austere morality of “rejection of the flesh.”
In addition, Julian attempted to unify all of the autonomous pagan temples into a structure similar to the Church. If Julian had succeeded in his grand attempt to revive paganism, it would have been as a state-sponsored institutional structure very similar to Catholicism in everything from its views on sexuality to its relationship with central authority. This is arguably reflected in the Julian Society, which seems to aim to reject the “counter-culture” associations of most of modern neo-paganism in favor of a “respectable” image as a serious and sober religious option in the modern world.