Relations between the Atiyah-Patodi-Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link L as the rho invariant of some closed three-manifold Y-L intrinsically associated with L. We study then the rho invariant of the manifolds obtained by the Dehn surgery on L along integer and rational framings. Inspired by the results of Casson and Cordon and Cimasoni and Florens, we give formulas expressing this value as a sum of the multivariable signature of L and some easy-to-compute extra terms.