It seems natural for me to put a shield in my right hand and a weapon in my left. No, I've never used any of this equipment IRL, but similar to snowboarding or skateboarding, people often have a preference for which foot would go first even before they actually attempt it. It just feels right. Some are easily able to shift from one to the other, and I don't have too much trouble playing a right-handed game since I've grown up in a majority right-handed world, but it's still a little awkward after a few hundred hours of gameplay.
I've recently started a spellsword-type character with a sword in my left hand and a spell in the right. Again, feels natural. This is intended to take advantage of the damage scaling of 1-Hand to augment the lack of damage scaling in destruction. Say what you will about the wisdom of it, but it's definitely a fun style so far. Now, I would like to apply a poison to the only physical weapon I have equipped, but I am unable to do so. I would have to un-equip, re-equip to my right hand, apply the potion, and switch back. I've tried it once or twice and it's a good way to temporarily ruin the normally excellent immersive gameplay. I wasn't going much into the alchemy skill anyway, but I wouldn't mind applying some of the poisons I find out and about in a more user-friendly manner. I'm also unable to equip a shield in my right hand, as I mentioned, with a spell in my left for the more difficult battles.
Finally, I'm not sure, but I've read that the attack speed of a left-hand equipped, single-handed weapon is slightly slower than a right-hand equipped, single-handed weapon. Seriously? I may be misinformed on this, but why would the developers intentionally insert an additional disadvantage for left-handed players (other than the obvious no-shield-in-right-hand)?

