When I was having a bad time with my Hennig sliding forward, the saddle fitter told me crossing the billets would help prevent the slippage. Unfortunately, it makes tightening the girth from the saddle darn near impossible. I eventually went to a Le Tixerant girth which was the only thing I ever tried (& I tried a lot) that prevented the saddle from moving. Interestingly, my new Stubben Genesis doesn't slide forward at all. Go figure.
Be careful doing this unless you KNOW your saddle is fitted correctly. The front billet crossing to the back can jam the front of the saddle/pommel down tightly. It may not shift anymore, but it may cause problems down the road. Just my experience.
PS...I cross my billets after a saddler very carefully showed me how and what to look for to be sure it was right on that particular horse.
Your beliefs don't make you a better person, your behaviour does.