For classical marriage markets with equal numbers of men and women and where all men find all women acceptable and all women find all men acceptable, Sasaki and Toda (1992) characterized the core by same-side anonymity for marriage markets, Pareto optimality, consistency, and converse consistency. Nizamogullari and Özkal-Sanver (2014) generalized this result to the domain of classical marriage markets by adding individual rationality and replacing same-side anonymity for marriage markets with a stronger property called gender fairness. We generalize both results by characterizing the core on the domain of solvable roommate markets without so-called “3-rings” (and on the domain of marriage markets) by individual rationality, anonymity, Pareto optimality, consistency, and converse consistency. We also prove that extending this characterization to the domain of solvable roommate markets is not possible.