We investigate modular properties of term rewriting systems, the basic operational formalism for equational specifications. First we study sufficient conditions for the preservation of the termination property under disjoint (and more general) combinations of term rewriting systems. By means of a refined analysis of existing approaches we show how to prove several new asymmetric preservation results. For this purpose we introduce two interesting new properties of term rewriting systems related to collapsing reductions: uniquely collapsing and collapsing confluent. We discuss these properties w.r.t. well-known confluence, consistency and normal form properties, and show that they are modular for left-linear systems, but not in general.