It turns out the equality between two boosts and a rotation followed or preceded by a single boost is correct: the rotation of frames matches the angular separation of the composite velocities, and explains how one composite velocity applies to one frame, while the other applies to the rotated frame. The rotation also breaks the symmetry in the overall Lorentz transformation making it nonsymmetric. For this specific rotation, let the angle be ε and the axis be defined by the unit vector e, so the axis-angle vector is ε = ε e.