By starting with an initial tetrahedron with some radius and affixing tetrahedra around both sides, the unique-valley shapes with an odd number of peaks are realized! The shapes with an even number of peaks just have a different initial configuration.

In this GIF, I've varied the initial radius from 0 to 10, adding new tetrahedra as they fit.

As the number of peaks/valleys grows, the shapes become more circular, and the defining angles become more complicated. For the unique-valley shape with 10 valleys, the inter-peak angle can be expressed as the root of a 12th-degree polynomial.

I wonder if there's a closed-form expression for this shape’s unique dihedral angle!

I call the edges that lie in the "upper" plane the "peaks”, and the concave edges that lie between the peaks the “valleys". For these "unique-valley" shapes, one valley has a non-right dihedral angle Ψ, while each other dihedral angle is right.

For the unique-valley shape with 5 peaks/valleys, the unique angle is around 166.06º. I've 3D-printed a version of this shape with a few top layers of black filament to accentuate the peaks and have attached a diagram of the shapes downward projection.