Capturing vertically in portrait mode eliminates the problem created by anastigmatic correction in lens.
A simple lens, like the eye, focuses on a spherical plane, which is why eyes evolved to be round not cubes and early cameras curved the film. So do specialty pano cameras like the WideLux.
When the focal plane flattened a simple lens will only be sharp in the center (astigmatism) with the rest of the image wi \"front focused\". To get everything in focus compound lenses use additional corrective (anastigmatic) elements to alter the path. By way of analogy its like cutting a tennis ball in half, nailing it to a board and the single point in the middle where it touches, them pulling and nailing the equator of the ball flat on the board.
As focal length gets shorter the edge distortion from the corrective elements is noticed more on the wide dimension of the image. Turning the camera vertical puts the distortion at the very bottom in the foreground and up in the sky in landscapes where is will not be noticed as much and minimizes the distortion across the image where the frames are joined.
So given the choice between shooting with a 20mm or a 50mm the better choice would be the 50mm because there would be less anastigmatic distortion seen with the 50mm. When the lens has any anastigmatic distortion shooting in portrait mode will minimize its visible impact. In either case you need to shoot and stitch more frames, but the net result is less distortion. Having a pano head that rotates the camera around the nodal point of the lens (where the image inverts) also minimizes parallax distortion.
