It seems to me there is a duality between linear mechanics and angular mechanics. Uniform motion along the circumference of a circle can be represented as a superposition of two perpendicular harmonic oscillations. Conversely, linear harmonic oscillation can be represented as a superposition of two counterrotating circular motions. (For instance, the motion of the bob of a Foucault pendulum can be represented as a superposition of solutions for the motion of a conical pendulum. In the mathematical solution the counterclockwise and clockwise motion have a different period, which corresponds to veering of the plane of swing.)
The following diagram expresses Newton's first law, and it also expresses an area law. Newton used that area law for his derivation of Kepler's law of areas from first principles.
In the absence of any net force an object will move along a straight line (ABCDE), covering equal distances in equal intervals of time.
Also, with respect to point 'S' the object sweeps out equal areas in equal intervals of time.
In terms of newtonian mechanics:
In order for a linear momentum to exist a single spatial dimension is sufficient, and motion in three spatial dimension can be represented as a superposition of three motions, one for each spatial dimension.
In order for an angular momentum to exist the minimum number of spatial dimensions is two, of course. Geometrically angular momentum corresponds to area.
In terms of newtonian mechanics angular momentum cannot be defined in the absence of any force. The absolute minimum configuration is two point masses that exert a force upon each other, thus each causing change of motion of the other. Then we can pinpoint a single point that we know to be an unaccelerated point: the Common Center of Mass of the two point masses. Generalizing: for angular momentum the common center of mass of all the participating masses is the reference of motion. Only the angular momentum with respect to the common center of mass is consistent.
(Beyond the scope of newtonian mechanics: it may be that a theory of motion is possible in which translational/rotational duality is complete. I don't know.)
As far as newtonian dynamics is concerned: I prefer to think in terms of some form of duality of linear and angular mechanics, so that the two are not independent.
Also, I concur with the statement by contributor Vadim about Newton's first law. I concur that in its historical form the first law is redundant.
With the benefit of hindsight we know that in any theory of motion (Newtonian, SR, GR) the first thing to assert must concern the geometric properties of the arena in which physics is taking place.
Reversing the historical sequence of theories:
GR: the metric of spacetime is described by a solution of the Einstein Field Equations
SR: the metric of spacetime is the Minkowski metric
Newtonian: space has the same symmetries as Euclidean geometry.
In the Principia we see that Newton immediately asserts that it is valid to do vector addition (back then it wasn't called 'vector addition' yet, but it is what Newton was doing/applying.) That is, in order to formulate newtonian mechanics it must be granted that Euclidean geometry is a valid model for the geometric properties of physical space.
In terms of Euclidean geometry symmetry under rotation is implicit.