When setting up the sphere, I accidentally stumbled into and solved two problems from Dorrie's 100 Great Problems of Elementary Mathematics:Ellipse from conjugate radii.
Ellipse inscribed in a parallelogram.

The parametric equation gives the direction of increasing t from a → b (Draw the segment AB, and follow it from A to B: that way).
Now consider point C, with radius c. Each of the possible pairs (±c, ±d) is a pair of conjugate radii. I take the principal conjugate following the direction of increasing t. This is an assumption which describes the problem I wish to solve. Direction remains a free choice, to be resolved in the context of the problem at hand. Like this:
In the Sphere problem, I adopted a right-handed, rectangular coordinate system. To begin, I manipulate the trihedron by allowing each axis to slide along one of the meridians through the coordinate axes. Naming the axes in alphabetical order, I can distinguish the correct conjugate:
e.g. on the ellipse determined by x, the vectors y and z are conjugate. Choose y at pleasure on the ellipse, and z is the direction of positive rotation. When the user manipulates the figure, I test the signs of the cross products among the axes to determine whether an axis has flipped sides front/back. z can always be uniquely assigned.