We demonstrate that the singular binding mechanism characterizing isoelectronic centers formed from two isoelectronic traps can also bind, in addition to the well-studied excitons, various number of charges. Using the emission fine structure of Te dyads in ZnSe and N dyads in GaAs, we establish that these pseudodonors and pseudoacceptors can bind positively and negatively charged excitons, respectively, and that both can bind biexcitons. This ability to bind various charge configurations, in addition to their very low inhomogeneous broadenings and perfectly defined symmetries, further establishes isoelectronic centers as an interesting alternative to epitaxial quantum dots for a number of applications.