Ambidextrous people may come by this naturally. I was writing with both hands still in 2nd grade, and the teacher rubber-banded the pencil to my right hand.
Switch hit, throw left, fly rod and bait rod left, spin rod right - that's how they came from the store.
It's probably noteworthy that spinning tackle rewards jerk in your cast, while both fly rod and bait rod react adversely to jerk (tailing loops and backlash).
They all go back to teenage years for me, when our pituitary is still producing learning hormones, and muscle memory comes easy. I was making jerk-free spiral casts with Ambassadeur for weightless free-shrimp within months on a baitcaster, and could double anyone's spinning distance casting the same weightless rig on Ambassadeur.
I have natural haul, and what most people don't get about fly rod, is your line hand should do more reading and loading than your rod hand - everything you do with your rod hand should be short and smooth. Probably a skill that comes more natural to southpaw casters. (S Texas instructor and guide Lefty Ray Chapa comes to mind)
I've been fishing fly rods in the dark most of my life, can make a vertical back-cast backed up to a cliff, change-direction mid-cast, always cast across my chest so there's no difference in forward cast or back-cast - I can present either way.
I can make an underhand cast that's akin to skipping a baitcaster to get under cypress overhang.
Best I could do for a cypress overhang photo, but can think of a spot where it covers 70% of the river channel, your underhand back-cast needs to be parallel on the narrow channel, and change direction on forward to sight-fish the bass.
I normally don't false cast, but begin with a roll pick-up, a single back-cast for line speed, and shoot. The exception to that is fishing beyond 70', and will make 2 false casts to get enough line speed to shoot my Teeny line to 140' (including that much backing, etc.) Of course on a shotgun cast to 70' on the 3rd stroke, beginning with a few feet of fly line, need a false cast to get through the belly.
The time in Alaska I used a guide's LHW fly reel, though, I didn't have the crank speed to keep up with a charging Kenai rainbow, so that muscle memory was never allowed to develop. I wouldn't have that problem on RHW.
My daughter can only tolerate cranking a reel with her right hand, so that's how we set her up on all tackle
Noteworthy, she doesn't care which arm she casts with and randomly switches.