The two un- words applied to Mulligan at the beginning of Telemachus emphasize his very un-priestly nature as he imitates the actions of a priest. His dressinggown is "ungirdled," and his hair is "untonsured."
The gown is ungirdled simply because Mulligan has not tied the cloth belt around his waist. But a priest's vestments are secured with a similar cloth band with tassels on the ends, called a cincture, and as the priest puts on the cincture he recites a prayer with the line, “Gird me, o Lord, with the girdle of purity.” Gifford plausibly infers that a dressinggown lacking the girdle of purity “suggests violation of the priestly vow of chastity.” Like his real-life counterpart Gogarty, Mulligan is an enthusiastic brothel-goer.
"Tonsure" derives from a Latin verb meaning to shear or shave. It refers to the part of a priest's or monk's head that is shaved, often just prior to taking holy orders. In Proteus Stephen thinks of priests "moving burly in their albs, tonsured and oiled and gelded." (The "alb" is the priest's white vestment, from the Latin word for white, alba.) Here again, the un-word applied to Mulligan suggests lack of reverence for the rite he is so blasphemously imitating.