This article is concerned with the strong stability problem of linear continuous-time delay-difference equations with multiple time delays. A family of linear matrix inequalities (LMIs), indexed by a positive integer $k$, is derived to assess strong stability. A time-domain interpretation of the proposed LMI-based condition is given in terms of a quadratic integral Lyapunov functional, which allows us to reveal relations with an existing result. The LMI condition can easily be reformulated in a form where the dependence on the coefficients of the delay-difference equation is affine, which is instrumental to establishing a sufficient LMI condition for robust strong stability of delay-difference equations with norm-bounded uncertainty. A necessary and sufficient condition for robust strong stability is also given, in the form of a structured singular value characterization. Examples are given to illustrate the effectiveness of the presented results.