Looking at the unit of action (kg m^2/s), it is the same as angular momentum, like Planck's constant. Usually action is seen as the time integral of energy, or sometimes as the spatial integral of momentum, but it also has the same unit as a time derivative of moment of inertia, or you can take the time integral of force to get momentum then the spatial integral of momentum to get angular momentum. Action also has the same units as the double spatial integral of mass flow (kg/s), or even the triple spatial integral of viscosity. I'm not sure if anybody integrates over charge, but the double charge integral of electric resistance or the charge integral of magnetic flux also have the same units as action. You could even integrate kinematic viscosity (area/s) over mass and get the units of action.
Action, like so many important quantities, has a factor of exactly length^2. So it looks like a spatial bivector sort of thing, and I have been told Lie algebras are basically bivector algebras. Other unit types that also have m^2 as an exact factor are: area, kinematic viscosity (area/s), moment of inertia, energy, power, magnetic flux, electric potential, inductance, resistance, and elastance (inverse capacitance). [Edit: Probability current has dimensions of 1/(m^2 s) and current density is in coulombs/(m^2 s). ]
Expressing action as "the time rate of change of moment of inertia" seems to me like the closest to being intelligible; minimizing the rate at which moment of inertia changes seems like a plausible rule.