In what specific orientation should the finished omelette land on the plate after inversion from the pan?

Seam-side down.

The final step in plating an omelette involves inverting the carefully shaped egg mass from the pan onto the plate. The goal of this maneuver is to ensure the most visually appealing and structurally sound surface faces upward. Since the folding process (whether rolling or half-moon folding) creates a seam where the edges meet, this seam represents the weakest or least attractive part of the exterior surface. Therefore, the professional technique dictates that the pan must be tilted so the omelette rolls or slides out such that the seam rests directly against the plate surface, leaving the smooth, unbroken, cooked exterior facing upward for presentation.