The present paper identifies unique symmetry properties of the d\ensuremath{\rightarrow}+d\ensuremath{\rightarrow}d+p+n breakup reaction that make it an excellent probe for studying charge-symmetry breaking. Measurements were made for two configurations of the ejected particles in the breakup reaction to obtain values of the spin observables ${\mathit{A}}_{\mathit{y}}$, ${\mathit{A}}_{\mathit{y}\mathit{y}}$, and ${\mathit{A}}_{\mathit{z}\mathit{z}}$. These observables are compared for the mirror reactions $^{2}\mathrm{H}$(d\ensuremath{\rightarrow},dp)n and $^{2}\mathrm{H}$(d\ensuremath{\rightarrow},dn)p for the two angle pairs (${\mathrm{\ensuremath{\theta}}}_{\mathit{d}}$,${\mathrm{\ensuremath{\varphi}}}_{\mathit{d}}$,${\mathrm{\ensuremath{\theta}}}_{\mathit{N}}$,${\mathrm{\ensuremath{\varphi}}}_{\mathit{N}}$)=(17.0\ifmmode^\circ\else\textdegree\fi{},0\ifmmode^\circ\else\textdegree\fi{},17.0\ifmmode^\circ\else\textdegree\fi{},180\ifmmode^\circ\else\textdegree\fi{}) and (17.0\ifmmode^\circ\else\textdegree\fi{},0\ifmmode^\circ\else\textdegree\fi{},34.5\ifmmode^\circ\else\textdegree\fi{},180\ifmmode^\circ\else\textdegree\fi{}) for an incident deutron energy of 12 MeV. In addition, spin observables for the $^{2}\mathrm{H}$(d\ensuremath{\rightarrow},pn)d reaction at ${\mathrm{\ensuremath{\theta}}}_{\mathit{p}}$=${\mathrm{\ensuremath{\theta}}}_{\mathit{n}}$ and ${\mathrm{\ensuremath{\varphi}}}_{\mathit{p}}$=${\mathrm{\ensuremath{\varphi}}}_{\mathit{n}}$+180\ifmmode^\circ\else\textdegree\fi{} are shown to provide a particularly good test of charge symmetry. Our ${\mathit{A}}_{\mathit{y}}$, ${\mathit{A}}_{\mathit{y}\mathit{y}}$, and ${\mathit{A}}_{\mathit{z}\mathit{z}}$ data for the $^{2}\mathrm{H}$(d\ensuremath{\rightarrow},pn)d reaction at (${\mathrm{\ensuremath{\theta}}}_{\mathit{p}}$,${\mathrm{\ensuremath{\varphi}}}_{\mathit{p}}$,${\mathrm{\ensuremath{\theta}}}_{\mathit{n}}$,${\mathrm{\ensuremath{\varphi}}}_{\mathit{n}}$)=(17.0\ifmmode^\circ\else\textdegree\fi{},0\ifmmode^\circ\else\textdegree\fi{},17.0\ifmmode^\circ\else\textdegree\fi{},180\ifmmode^\circ\else\textdegree\fi{}) are used to illustrate this latter point. Of the ten charge-symmetric sets of observables measured, two were found to differ by 2.5 standard deviations.