I have seen a couple replies with the thought of backing the horse after mounting.
I strongly suggest not to back the horse after mounting. I have seen and know of several people seriously injured by horses flipping over backwards from being backed after mounting. I am not saying that every horse will flip over but if you ride a lot of young horses or unfamiliar ones I wouldn't do it. I had a boss that would fire you on the spot if he caught you doing it.
I may come from a little more old school style of doing things, but I would much rather have my left rein a tad bit shorter if by chance he moves off then at least he is circling into me rather than away. If he moves off to the right then there is the possibility of him knocking me down with a hip and cow kicking me on the way out. Also If a horse decides to buck after I get on, at least with a shorter inside rein, I have a better chance of talking him out of it and getting control. And I am setup to untrack him to the left rather than riding straight off.
I will also bump a rein and reinforce whoa when getting on. I still keep my toe in the stirrup, not my whole foot. With just my toes in the stirrup I can get out if I have to and reduce the chance of getting hung up should something happen. And with just the toe in the stirrup I reduce the possibility of jabbing my horse in the ribs as I swing up.
I was taught not to grab the cantle board when mounting. This also will pull the saddle off to the side. I stand at the shoulder, grab some mane on the neck with my rein hand, the right hand goes over the top of my saddle to the offside under my rope and grabs the bucking roll, or swell, toe in stirrup and swing up and smoothly as possible.
I DON'T LEAD 'EM AND FEED 'EM, I RIDE 'EM AND SLIDE 'EM.