When Mulligan talks in Telemachus of his name having "a Hellenic ring," and again when he says that he and Stephen working together "might do something for the island. Hellenise it," he is voicing a tenet of late Victorian counterculture advanced by the poet and man of letters "Matthew Arnold" (1822-1888). Arnold's theory informs statements elsewhere in Ulysses about Greekness and Jewishness.
Arnold coined the verb “to Hellenise" in the fourth chapter of Culture and Anarchy (1869), which distinguishes the moral impulse of Hebraism (letting revealed truth guide one's actions) from the intellectual impulse of Hellenism (seeking truth disinterestedly, using human rather than divine guides). Arnold felt that English culture was excessively Hebraic and needed rebalancing.
By the end of the century, however, Arnold’s terms had been co-opted by the avant-garde movement and made considerably more prejudicial: “Greek,” Gifford observes, now connoted bohemian freedom, sensual pleasure, and aesthetic beauty, while “Jew” connoted social repression, “straightlaced Victorian morality,” and hostility to art. Mulligan embodies the decadence of this fin-de-siècle scene, and he shows none of Arnold’s interest in restoring cultural balance. His Hellenism consists of Swinburnean values: flashy style over intellectual substance, and hedonism over conventional morality. In Scylla and Charybdis he praises "The Greek mouth that has never been twisted in prayer."
Ulysses also drops a hint or two that Mulligan may entertain the ancient Greek openness to choice of genders in sexual pursuits. In the same part of Scylla and Charybdis, having seen Bloom looking between the nether cheeks of the statue of a Greek goddess, he not only infers that the Jew is interested in a forbidden kind of intercourse: “O, I fear me, he is Greeker than the Greeks. His pale Galilean eyes were upon her mesial groove.” He also infers that such interest implies sexual ambidexterity: “The wandering jew . . . Did you see his eye? He looked upon you to lust after you. I fear thee, ancient mariner. O, Kinch, thou art in peril. Get thee a breechpad.” A certain amount of projection seems to be going on. Mulligan sounds distinctly queeny in recounting what Dowden said about Shakespeare: “Lovely! Buck Mulligan suspired amorously. I asked him what he thought of the charge of pederasty brought against the bard. He lifted his hands and said: All we can say is that life ran very high in those days. Lovely!” To which someone in the following line (almost certainly Stephen) thinks, “Catamite.”
Mulligan’s joking about Bloom being simultaneously a Jew and Greek points comically and obliquely (the typical way in which this book makes meaning) toward the Arnoldian idea that (contra Mulligan’s own practice) these two cultural impulses might coexist, combine, and even coincide. In Circe a cap exclaims, “Jewgreek is greekjew. Extremes meet. Death is the highest form of life. Bah!” This coincidence of contraries is manifested most obviously in the coming together of the book’s two principal characters. The first chapter identifies Stephen with Hellenism in ways ranging from the accident of his “ancient Greek” names to his principled refusal to compromise his intellect for the sake of his mother’s piety. Bloom will be identified with the ancient, archetypal figure of the wandering Jew, and his relentlessly practical and moral sensibility contrasts starkly with Stephen’s high intellectual and aesthetic disregard for such things (even though he has, like Stephen, rejected religion). To the extent that some uncanny Father-Son unity is realized between these very different men, jewgreek really is greekjew.