For any given cross polytope, having vertices described by mutually perpendicular vectors $\{\pm a_1, \pm a_2,\dots,\pm a_n\}$ defined on $\mathbb{R}^n$, it is known that an orthogonal projection onto any subspace will produce a eutactic star. Are there any conditions or scenarios where the angles between the corresponding points of the star will be constrained or have some symmetry?
My intuition is that this is dependent on the nature of the projection - any orthogonal projection will produce a star but perhaps not one that exhibits a constant angle between neighbouring points?