As I see it, given my understanding of both disciplines.
In a dressage figure 8, two perfect circles touch at one spot. The horse bends along the line of the circles. In an ideal world, the horse takes one straight stride directly over the center line while switching from one bend to the other. Ring geometry will indicate exactly where the figure should occur and will insure that the circles match. Performed down the length of the arena.
In a saddle seat figure 8,
each half is closer to a D than to a O. Take several strides down the “landing strip” in the center. On the curves, don’t go all the way to the rail. This takes too much time and annoys the judge. The second loop should match the first. Come back to the same place. A second figure 8 should follow the footsteps of the first. Performed across the width of the arena.
In a dressage serpentine, three half circles meet on the center line. Bend. One straight stride at center point. Follow ring geometry.
In a saddle seat serpentine, the loops are closer to u-shapes. Make sensible use of the arena. All loops should match.
Experienced saddle seat/dressage riders, yeah or nay?
My advisors (comment below with corroborating experts) tell me this is a better representation of a saddle seat figure 8. The “double capital D” comment that I have heard is an aide memoire to keep riders from going off on a diagonal, thereby creating a giant X in the center. Not a desirable figure in dressage either. “I was taught to try to get it round, and 3 strides straight max. So it would be more of a two oval situation than your dressage loops.” Thank you Ms. Bourbon.