In this paper, some geometric aspects of indistinguishability operators are studied by using the concept of morphism between them. Among all possible types of morphisms, the paper is focused on the following cases: Maps that transform a T-indistinguishability operator into another of such operators with respect to the same t-norm T and maps that transform a T-indistinguishability operator into another one of such operators with respect to a different t-norm T ′. The group of isometries of a given T-indistinguishability operator is also studied and it is determined for the case of one-dimensional operators, in particular for the natural indistinguishability operators E T on [0, 1]. Finally, the indistinguishability operators invariant under translations on the real line are characterized.

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