What I can say is, my left hand is important to me, beyond being useful.
At one time I wondered, if I needed to put my hand over my heart, whether I should somehow use my left hand. I found a way to do so: right hand, over my left hand, over my heart. And I looked down at that, both of my hands on my chest, both feeling my heartbeat, and I thought: That could work. That seems right for me.
Now when I put my hand over my heart for the flag and for the National Anthem, I put both hands.
I actually do this in private. I most often see the U.S. flag and hear "The Star-Spangled Banner" at sporting events, which I almost always see on television and rarely in person...
(Speaking of "The Star-Spangled Banner," here's one thing my father and I agree on: we tend not to like non-traditional renditions of the National Anthem. Adding a lot of notes, jazzing it up, adding vocal effects: I can't answer for Dad, but to me, it doesn't add much to the song. It's a difficult song, relatively speaking, with almost too much range to sing comfortably and already not all that straightforward; remember, this song about a battle early in this nation's history is a questioning song, at least as sung (without its answering final stanzas), and in storytelling terms kind of leaves us hanging like the non-ending to the Fisher King myth -- and come to think of it, maybe the anthem's effect should be to make us think about what comes next, what can we do to get through the next adversities, what can we do to lessen said adversity, what can we do to make this country work better. The danger with tweaking the song too much is to make one think too much about just the song. But I've really digressed. Yes, plenty of liberals like me think about such stuff...)
...but still, earlier today, when the U.S. flag was presented at Soldier Field before the Packers-Bear NFC Championship Game and the National Anthem was sung, I once again put both hands over my heart.
What happens when I do that in person at an event? I probably will eventually. And I can explain my reasons: it's a personal way to observe the spirit of both the flag and the anthem, while giving my left hand some more support. Being left-handed is kind of important to me, after all. I've adapted decently to being left-handed in a part of the world that better fits right-handed people, and I appreciate that people are less likely to try and train left-handedness out of people, a dumb thing that was still happening plenty of times even a few decades ago (didn't Tom Cruise get it "trained out" of him as a kid? And he's relatively young!), so: I want the left hand to get some more love. More good attention.
And I'll hope it indeed fits the spirit of why we do the act.