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I just realized that Angular momentum and Action have the same units. The first one is usually defined as $\mathrm{kg\frac{m^2}{s}}$, but that is equal to $\mathrm{Js}$.

I know that Noerther's theorem says that if the action is rotationally symmetric then the system will exhibit angular momentum conservation. (If the action is time symmetric then energy is conserved, but energy and action don't have the same SI units).

Does some other (more direct or more abstract) connection between these two quantities exist, apart from having the same units?

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Action and angular momentum have the same dimensions due to an accident of dimensional analysis.

By definition, the Lagrangian has dimensions of action per time. Given a generalized coordinate $\xi$, the corresponding conjugate momentum is given by $\partial L/\partial \dot \xi$ and therefore has dimensions of action divided by the dimension of $\xi$.

From there it follows that angular momentum has dimensions of action because it's conjugate angular coordinate is dimensionless.

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