The saddle would best resemble english riding. . .a small light pad mounted where neck joins torso, the stirrups hang to either side of the neck, feet in from behind them.
Two bronze rings have been set into the corners of the beaks, with leather loops.
Usual riding position is in the saddle, all the way leaned forward, holding the loops. (your torso flat against the back of the bird's neck)
Pulling left and right controls direction, pulling up or down results in climbing, diving or banking.
Took a lot to get used to it, and you've all tried riding sitting up on a soar, but the speeds are a lot faster, you doubt you could ride in that position for anything other than soaring. (So far, none of you have successfully made an attack on anything from bird back, except dropping a stone rather innacurately.)