Given the topology of a plane in 2D, it's not precise that polar coordinates accept multiple angles. Polar coordinates that span more than 360 degrees (2 pi) describe something else than the Cartesian plane.

For example, if I attach a rope to the origin on the plane and I rotate the rope more than once (increase theta), it will not have the same "position" than if I only limit the rotation to 360 degrees. The rope will start to roll onto itself, thus it undergoes a different topology.

Essentially, beyond 360 degrees, the angle is no longer a coordinate of the orthogonal coordinate system. One cannot achieve such a rolled up position of the rope onto itself in a Cartesian coordinate system by selecting any x,y point to position the rope. One cannot equal apples to oranges.

An equivalent topology would be to pile up multiple Cartesian coordinate spaces on top of each other to compare with polar system with angle beyond the 360 deg range.

Given that r is a polar coordinate, r cannot be negative, otherwise it would mean that the polar coordinate system would not be orthogonal, which it is.

One cannot adopt a convention that contradicts the definition of "coordinates" and the fact that for orthogonal coordinate systems each coordinate do not depend on any other coordinate, otherwise, it would mean that the coordinate system is not orthogonal.

For this reason, negative r is not a coordinate of a polar coordinate systems, since it needs pi in the angle coordinate, expressed as -r = (r, pi) and that the polar coordinate system is an orthogonal coordinate system.

Calling -r a coordinate is a travesty.