[en] I am pleased to be able to attend this symposium in honor of D. Allan Bromley and to see the new accelerator of the Yale University Nuclear Structure Laboratory. My talk on symmetry tests seems appropriate for this occasion: so much of the progress in this field depends on detailed knowledge of nuclear structure. The nuclear ''tricks'' that are played to filter and amplify interactions are possible because the nuclear spectroscopists have cataloged nuclear levels and determined their properties. I will describe how such nuclear structure studies may help to provide a window on physics beyond the standard model. My talk is not a summary of this subfield of nuclear physics. There is simply too much happening today to make a summary talk feasible. Instead, I have chosen four topics that I hope are representative of the field as a whole: parity mixing of nuclear states, time-reversal-odd nuclear moments, the Mikheyev-Smirnov enhancement of solar neutrino oscillations, and a nuclear experiment to monitor the long-term rate of stellar collapse in the galaxy. 39 refs., 5 figs., 1 tab