= Solution
The two continuous, time-independent <symmetries> are time translation and a common <rotation> of both angles. Their <Noether charges> are respectively the total <energy> and vertical <angular momentum>:
$$
\boxed{E=\frac12I_1\dot\theta^2+\frac12I_2\dot\varphi^2+\frac k2[a^2+A^2+h^2-2aA\cos(\theta-\varphi)],}
$$
$$
\boxed{J=I_1\dot\theta+I_2\dot\varphi.}
$$
For the second symmetry, $\theta\mapsto\theta+\epsilon$ and $\varphi\mapsto\varphi+\epsilon$, so the <Noether theorem> gives the sum of their conjugate momenta. Independent <rotations> are not <symmetries> of the spring coupling.
There are also <discrete symmetries>: simultaneous reflection $(\theta,\varphi)\mapsto(-\theta,-\varphi)$, and <time reversal in classical mechanics>, which reverses both angular velocities. These leave the dynamics invariant but provide no additional infinitesimal <Noether charges>. Combined with common rotations, the reflections give the full planar orthogonal symmetry of the angular geometry. If $I_1=I_2$, interchange of the two angular coordinates is an additional <discrete symmetry> of the reduced dynamics.
There is a further useful time-dependent <symmetry>, up to a <total derivative>, on a continuous lift of the angles: adding the same constant <angular velocity>, $(\theta,\varphi)\mapsto(\theta+vt,\varphi+vt)$. It changes $L$ by
$$
\frac d{dt}\left[v(I_1\theta+I_2\varphi)+\frac12(I_1+I_2)v^2t\right].
$$
Its conserved quantity is $\boxed{I_1\theta+I_2\varphi-tJ}$. It is the integration constant for free common rotation, rather than a new globally single-valued angle charge on the configuration torus. The globally defined continuous charges are $E$ and $J$ above.
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