Direction is currently considered to be a ratio between lengths.  This is different (mathematically and conceptually) from representing direction as a quantity.  Ratios are not quantities.   They are a comparison between two or more quantities, which is fundamentally different from how quantities themselves are represented.

The current state of the art allows for direction to be expressed as a ratio.  It is not really a quantity by today’s standards.  The two basic methods are to consider a ratio between two (or more) perpendicular lengths (Pythagoras) or to consider a ratio between the lengths of the diameter and circumference of a circle ( $\pi$).  In either case, these lengths are combined in a specific way in order to use them to represent a distinction that can be made between two separate directions.  This value is always a ratio, never a quantity.

The way it’s currently done, the radian is the cheater’s way of representing a turn.  It is conceptually a two-dimensional construct, and is only an abstraction of what really occurs with an actual physical turn in spacetime.  A turn in Euclidean 3-space is different than simply a ratio between two lengths.  The same problem exists with representing volume as a product of three lengths.  This assumes that the three lengths are in three directions that are “orthogonal” to one another.  What does that even mean without a base quantity of direction?  Orthogonal, but without a unit of direction?  What’s going on here?