The Levine–Tristram signature admits a µ-variable extension for µ-component links: it was first defined as an integer-valued function on $(S^1\setminus\{1\})^\mu$, and recently extended to the full torus $\mathbb{T}^\mu$. The aim of the present article is to study and use this extended signature. Firstly, we show that it is constant on the connected components of the complement of the zero locus of some renormalized Alexander polynomial. Then, we prove that the extended signature is a concordance invariant on an explicit dense subset of $\mathbb{T}^\mu$. Finally, as an application, we present an infinite family of three-component links with the following property: these links are not concordant to their mirror image, a fact that can be detected neither by the non-extended signatures, nor by the multivariable Alexander polynomial, nor by the Milnor triple linking number.

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 Gordon 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.