  We study magnetic dipole (M1) transitions between two quarkonia in the
framework of non-relativistic effective field theories of QCD. Relativistic
corrections of relative order v^2 are investigated in a systematic fashion.
Non-perturbative corrections due to color-octet effects are considered for the
first time and shown to vanish at leading order. Exact, all order expressions
for the relevant 1/m and 1/m^2 operators are derived. The results allow us to
scrutinize several potential model claims. In particular, we show that QCD
excludes both contributions to the anomalous magnetic moment of the quarkonium
induced by low-energy fluctuations and contributions to the magnetic dipole
operators of the type induced by a scalar potential. Finally, we apply our
results to the transitions J/psi -> eta_c gamma, Upsilon(1S) -> eta_b gamma,
Upsilon(2S) -> eta_b(2S) gamma, Upsilon(2S) -> eta_b gamma, eta_b(2S) ->
Upsilon(1S) gamma, h_b(1P) -> chi_{b0,1}(1P) gamma and chi_{b2}(1P) -> h_b(1P)
gamma by assuming these quarkonium states in the weak-coupling regime. Our
analysis shows that the J/psi -> eta_c gamma width is consistent with a
weak-coupling treatment of the charmonium ground state, while such a treatment
for the hindered transition Upsilon(2S) -> eta_b gamma appears difficult to
accommodate within the CLEO III upper limit.
