  Using 24 million $\psi' \equiv \psi(2S)$ decays in CLEO-c, we have searched
for higher multipole admixtures in electric-dipole-dominated radiative
transitions in charmonia. We find good agreement between our data and
theoretical predictions for magnetic quadrupole (M2) amplitudes in the
transitions $\psi' \to \gamma \chi_{c1,2}$ and $\chi_{c1,2} \to \gamma J/\psi$,
in striking contrast to some previous measurements. Let $b_2^J$ and $a_2^J$
denote the normalized M2 amplitudes in the respective aforementioned decays,
where the superscript $J$ refers to the angular momentum of the $\chi_{cJ}$. By
performing unbinned maximum likelihood fits to full five-parameter angular
distributions, we determine the ratios $a_2^{J=1}/a_2^{J=2} =
0.67^{+0.19}_{-0.13}$ and $a_2^{J=1}/b_2^{J=1} = -2.27^{+0.57}_{-0.99}$, where
the theoretical predictions are independent of the charmed quark magnetic
moment and are $a_2^{J=1}/a_2^{J=2} = 0.676 \pm 0.071$ and $a_2^{J=1}/b_2^{J=1}
= -2.27 \pm 0.16$.
