My own mounting technique is a bit different from most. I need to hold on to the horse to balance myself so I can jump a little to get my left foot into the stirrup. I'm too stiff to just place it there. Then I hop to get 90 out, facing the horse directly. Staying as close to the side as I can, I then go up with my left hand on the far side of his withers, and my right on the far edge of the pommel.
My goal is to get up very quickly and place my shoulders past the midline of the horse. This balances my weight over the horse, similar to when I sit in the saddle. THEN I swing around to put my right leg over the horse.
When I do it right, there is only about a half second where my weight isn't centered over the horse's back. It seems to make the horses happier.
I will also sometimes practice with them, going up and then balancing there for 10 seconds or so. I don't throw my leg over, but just stand there with my left foot in the stirrup and my weight balanced over their backbone. I expect them to stand still the whole time. If they move, I simple lean back a little and put my right foot on the ground (and letting my left fall out of the stirrup), and now they have bought themselves some time backing up and disengaging.
I expect to be able to do that 4-5 times in a row, with some head rubs and wither scratches in between.
I was told by a couple of instructors that I should be facing forward, with my left leg in the stirrup but my right side against the horse as I go up - ain't happening for this 53 year old man! If you face forward before going up, your right leg is twisted over and not vertical. That robs it of strength.