Mounting on the near side or off side
The question is worth looking into. Emperor Frederick II devotes an entire chapter to it and astutely analyses the question. If the falcon, carried on the left hand, struggles when the rider puts his or her left foot in the left stirrup, it runs the risk of alighting on the horse's neck, clenching its claws and provoking violent reactions from the horse. When putting the right foot in the right stirrup (off side), the rider places his or her left hand on the seat of the saddle or on the cantle bag and so does not run this risk.
For the off side, with the falcon carried on the left hand, one can also consider allowing the game bag – such is valid for falconry on horseback alone – to hang on the right haunch and so the falconer is not bothered by it when sitting down.

Image :

Henri Desmonts and his horse Jim hunting