From Wikipedia, conjugate variables have a general definition:
In classical physics, the derivatives of action are conjugate variables to the quantity with respect to which one is differentiating. In quantum mechanics, these same pairs of variables are related by the Heisenberg uncertainty principle.
In the same way that the conjugate of linear momentum is position ($x$), the conjugate of angular momentum is "angular position", a.k.a. orientation. You can find a list of other conjugate pairs here.
As for the units of orientation, indeed, radians are dimensionless:
Although the radian is a unit of measure, it is a dimensionless quantity. This can be seen from the definition given earlier: the angle subtended at the centre of a circle, measured in radians, is equal to the ratio of the length of the enclosed arc to the length of the circle's radius. Since the units of measurement cancel, this ratio is dimensionless.