The angular velocity of a particle $\omega = r \times v$ is a pseudovector because it is formed by the cross product of two vectors (position and linear velocity).
Likewise the angular acceleration of a particle $\alpha = r \times a$ is a pseudovector.
This implies to me that the angular position $\theta$ is ALSO a pseudovector. The only way I knowing of telling whether a given quality is a pseudovector, though, is if it has a cross product representation.
Looking at the form of the cross products which define $\omega$ and $\alpha$ might momentarily lead one to believe that the product must be $\theta = r\times r$, until you realize that that would imply that $\theta$ is always $0$.
My question is is there a cross product representation of the angular position of a particle?
No, angular position is no vector, just a scalar. The closest vector description is via $\omega$, which is orthogonal to the plane of rotation, honouring the right hand rule.
$\omega$ is an axial vector, as it will not change if $r=(x,y,z) \to -r = (-x,-y,-z)$.