One of the most effective methods developed for the analysis of security protocols is an approach based on equational reasoning and unification. In this approach, it is important to have the capability to reason about the knowledge of an intruder. Two important measures of this knowledge, defined modulo equational theories, are deducibility and static equivalence. We present new combination techniques for the study of deducibility and static equivalence in unions of equational theories sharing constructors. Thanks to these techniques, we obtain new modularity results for the decidability of deducibility and static equivalence. In turn, this should allow for the analysis of protocols involving combined equational theories which previous disjoint combination methods could not address due to their non-disjoint axiomatization.