Theoretical calculations of exciton magnetic-polaron (EMP) energies in semimagnetic CdTe/${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Te symmetric quantum wells have been performed. The influence of a spontaneous symmetry breaking of the carrier's wave function (the so-called interface-bound magnetic polaron) on EMP energies is investigated for various magnetic (Mn) ion concentrations and valence-band offsets. It is demonstrated that the stability of interface-bound magnetic polarons in perfect quantum wells depends on the valence-band offset K, and that, for the accepted value of K in the CdTe/${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Mn}}_{\mathit{x}}$Te system, interface-bound magnetic polarons can form only in samples with low-x values. It is also shown that interface-bound magnetic polarons are destabilized by the presence of interface disorder, or as a result of applying a magnetic field or increasing the lattice temperature. \textcopyright{} 1996 The American Physical Society.