  We present a model-independent approach to electric quadrupole transitions of
deformed nuclei. Based on an effective theory for axially symmetric systems,
the leading interactions with electromagnetic fields enter as minimal couplings
to gauge potentials, while subleading corrections employ gauge-invariant
non-minimal couplings. This approach yields transition operators that are
consistent with the Hamiltonian, and the power counting of the effective theory
provides us with theoretical uncertainty estimates. We successfully test the
effective theory in homonuclear molecules that exhibit a large separation of
scales. For ground-state band transitions of rotational nuclei, the effective
theory describes data well within theoretical uncertainties at leading order.
In order to probe the theory at subleading order, data with higher precision
would be valuable. For transitional nuclei, next-to-leading order calculations
and the high-precision data are consistent within the theoretical uncertainty
estimates. We also study the faint inter-band transitions within the effective
theory and focus on the $E2$ transitions from the $0^+_2$ band (the "$\beta$
band") to the ground-state band. Here, the predictions from the effective
theory are consistent with data for several nuclei, thereby proposing a
solution to a long-standing challenge.
