  We examine conditions under which a pair of re-arrangement invariant function
spaces on $[0,1]$ or $[0,\infty)$ form a Calder\'on couple. A very general
criterion is developed to determine whether such a pair is a Calder\'on couple,
with numerous applications. We give, for example, a complete classification of
those spaces $X$ which form a Calder\'on couple with $L_{\infty}.$ We
specialize our results to Orlicz spaces and are able to give necessary and
sufficient conditions on an Orlicz function $F$ so that the pair
$(L_F,L_{\infty})$ forms a Calder\'on pair.
