In topology, universal objects often serve as models for an entire class of spaces. A space $X$ is called a universal domain in a class $\mathcal C$ if every member of $\mathcal C$ is a continuous image of $X$. Classical examples include the Cantor set among compact metrizable spaces and the arc among Peano continua. In this talk, we give a new notion, called a reciprocating domain. A space $X$ in a class $\mathcal C$ is a reciprocating domain if whenever another space $Y\in\mathcal C$ admits a continuous surjection onto $X$, then $X$ also admits a continuous surjection onto $Y$. Intuitively, such a space cannot be reached from a ``larger’’ space in the surjective order without also being able to map back onto that space.

We discuss general properties of reciprocating domains and explain their relationship with universal domains. The main part of the talk will focus on examples arising in continuum theory, including chainable continua, tree-like continua, solenoids, circle-like continua, fans, and several related classes. Along the way, we will see that some familiar universal continua are also reciprocating, while in other natural classes reciprocating domains do not exist at all.