The purpose of this note is to explore the link between the ergodicity of the cluster equivalence relation restricted to the infinite locus and the indistinguishability of infinite clusters. It is an important element of the dictionary connecting orbit equivalence and percolation theory. This paper starts with a short exposition (as self-contained as possible) of some standard material of these theories. Then, the classical correspondence between ergodicity and indistinguishability is presented. Finally, we introduce a notion of strong indistinguishability that corresponds to strong ergodicity, and obtain that this strong indistinguishability holds in the Bernoulli case.