  We compute the magnetic dipole transitions between low lying Heavy Quarkonium
states in a model independent way. We use the weak-coupling version of the
effective field theory named potential NRQCD with the static potential exactly
incorporated in the leading order Hamiltonian. The precision we reach is
$k_{\gamma}^3/m^2\times{\cal O}(\alpha_s^2,v^2)$ and
$k_{\gamma}^3/m^2\times{\cal O}(v^4)$ for the allowed and forbidden transitions
respectively. We also resum the large logarithms associated to the heavy quark
mass scale. The specific transitions considered in this paper are the
following: $\Upsilon(1S) \to \eta_b(1S)\,\gamma$, $J/\psi(1S) \to
\eta_c(1S)\,\gamma$, $h_b(1P) \to \chi_{b0,1}(1P)\,\gamma$, $\chi_{b2}(1P) \to
h_b(1P)\,\gamma$, $\Upsilon(2S) \to \eta_b(2S)\,\gamma$, $\Upsilon(2S) \to
\eta_b(1S)\,\gamma$ and $\eta_b(2S)\to\Upsilon(1S)\,\gamma$. The effect of the
new power counting is found to be large and the exact treatment of the soft
logarithms of the static potential makes the factorization scale dependence
much smaller. The convergence for the $b\bar b$ ground state is quite good, and
also quite reasonable for the $c\bar c$ ground state and the $b\bar b$ 1P
state. For all of them we give solid predictions. For the 2S decays the
situation is less conclusive, yet our results are perfectly consistent with
existing data, as the previous disagreement with experiment for the
$\Upsilon(2S) \to \eta_b(1S)\,\gamma$ decay fades away. We also profit to
compute some expectation values like the electromagnetic radius, r^2, or p^2.
We find r^2 to be nicely convergent in all cases, whereas the convergence of
p^2 is typically worse.
