I see this flickering too on FireFox, for like an instant (when going left).
It makes me thing about some kind of loading delay due to the auto-mirror, as if the "other side" animations wouldn't be available in memory before the very moment you try to display it.
So, what about, in 1-1 sheet, on start of layout, trying to set mario's angle to 180 and set through all the animations, before setting him finally to the "start" animation and angle ?
Does it "correct" things up ?
If it does, but flickers at start, for the time of the startup, maybe consider displaying him as an effect, or something to "hide" the flickering. Once it is ok, simply display as intended.