A dynamical symmetry under Galilean transformations. For an isolated pair with constant masses, a force law depending only on relative positions and relative velocities has this invariance. Equal-and-opposite forces alone do not imply it: their magnitude can depend on centre of mass velocity relative to a preferred frame.
For a Galilean transformation , , with constant rotation and boost , velocity transforms as and acceleration as . With invariant inertial mass, Newton's second law retains its form only if the actual interaction laws and their physical background data transform to give . Acceleration invariance alone does not prove covariance of an arbitrary force law. For example, resistance depends on velocity relative to a medium, whose velocity must transform too.
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A law of physics is Galilean invariant if the same formula works both when you are standing still on land, or when you are on a boat moving at constant velocity.
For example, if we were describing the movement of a point particle, the exact same formulas that predict the evolution of must also predict , even though of course both of those will have different values.
It would be extremely unsatisfactory if the formulas of the laws of physics did not obey Galilean invariance. Especially if you remember that Earth is travelling extremelly fast relative to the Sun. If there was no such invariance, that would mean for example that the laws of physics would be different in other planets that are moving at different speeds. That would be a strong sign that our laws of physics are not complete.
Lorentz invariance generalizes Galilean invariance to also account for special relativity, in which a more complicated invariant that also takes into account different times observed in different inertial frames of reference is also taken into account. But the fundamental desire for the Lorentz invariance of the laws of physics remains the same.