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A list of probable lefties
The vast majority of FE character art has them holding a weapon or gesturing or doing something else asymmetrical. From that, you can guess which FE characters are lefties. :D
Note: This list is not complete. Obviously you can't tell either way for characters whose art doesn't involve anything suggesting hand dominance. Also, archers' bow hands only suggest a dominant eye, not a dominant hand. (Fwiw, Neimi, Innes, Leonardo probably shoot left-dominant. Rolf holds his bow in his right hand in his art, but holds it in his left when preparing to make an actual shot in Execution.) Similarly, I discounted staff users leaning on their staves, as right-handed people often hold canes in their left hand.
Here's a list for FE7-10 and 13. I'd be interested to see lists for other games!
EDIT: Lances and large axes are dubious grounds for left-handedness; see discussion and Mark's post. Characters who are categorized based on a lance or axe are noted.
FE7
Lucius
Dart
Farina? (She could be focusing mostly on the money and merely "holding" her lance...)
Hawkeye? (axe)
Hector? (axe; holds it in his right in FE6)
Karla
Matthew
Nergal
Oswin? (lance)
Vaida? (lance)
FE8
Artur (Assuming tomes are held in the off hand so that the dominant hand is free to turn pages)
Lute
Kyle? (Leaning on his lance like a cane)
Orson
Marisa (Canon per Tethys supports)
Saleh? (Conflicts with CG; see Raphi's comment)
Ewan (It's a little ambiguous wtf his art is doing, but he seems to be gesturing with his left. Conflicts with Cg; see Raphi's comment)
Syrene (lance)
Knoll (based on CGs; see Raphi's comment)
Seth? (Sword in his left hand in a CG; see Raphi's comment)
FE10
Nealuchi? (Holds a cane in his right hand)
Sephiran (Staff user, but brandishing rather than leaning)
Haar? (Leaning on his axe, but that means the right hand would be higher on the axe when picking it up from that position, which would be the hand more in control if he uses both hands.)
Heather
Kieran (See FE9 art; probable since it's a small axe)
Kyza
Nephenee? (two-handed lance? held in left in both FE9 and FE10 art)
Oscar? (lance; left in FE10, right in FE9)
Rhys
Not Sothe (he holds a dagger in his left in his art for both games, but he's wielding it in his right in the RD cutscene)
Titania? (Her left hand is "higher" on the axe in her FE10 art, but the axe is in her right hand in her FE9 art)
Calill? (Holding books in right in FE9, but there are two and it may be heavy enough to warrant the dominant arm)
Despite my headcanon, Pelleas actually looks like a rightie.
FE13
Inigo
Miriel (She is literally holding a quill pen in her left hand, sooo.)
Vaike (axe)
Henry
Ricken
Validar
Small-framed Boy Robin (The in-game portraits are probably meant to be viewed facing-right, per comparison to Chrom's design, level-up default position, etc. That would make this avatar be holding a book in his right hand. This doesn't seem to absolutely hold, however -- Flavia's holding a sword in her right hand in both her official art and her left-facing portrait.)
DLC!Roy
DLC!Micaiah (but she doesn't seem to have joints so who knows if usual handedness rules apply)
DLC!Elincia? (two-handed lance?)
DLC!Eirika? (left hand higher on staff)
Note: This list is not complete. Obviously you can't tell either way for characters whose art doesn't involve anything suggesting hand dominance. Also, archers' bow hands only suggest a dominant eye, not a dominant hand. (Fwiw, Neimi, Innes, Leonardo probably shoot left-dominant. Rolf holds his bow in his right hand in his art, but holds it in his left when preparing to make an actual shot in Execution.) Similarly, I discounted staff users leaning on their staves, as right-handed people often hold canes in their left hand.
Here's a list for FE7-10 and 13. I'd be interested to see lists for other games!
EDIT: Lances and large axes are dubious grounds for left-handedness; see discussion and Mark's post. Characters who are categorized based on a lance or axe are noted.
FE7
Lucius
Dart
Farina? (She could be focusing mostly on the money and merely "holding" her lance...)
Hawkeye? (axe)
Hector? (axe; holds it in his right in FE6)
Karla
Matthew
Nergal
Oswin? (lance)
Vaida? (lance)
FE8
Artur (Assuming tomes are held in the off hand so that the dominant hand is free to turn pages)
Lute
Kyle? (Leaning on his lance like a cane)
Orson
Marisa (Canon per Tethys supports)
Saleh? (Conflicts with CG; see Raphi's comment)
Ewan (It's a little ambiguous wtf his art is doing, but he seems to be gesturing with his left. Conflicts with Cg; see Raphi's comment)
Syrene (lance)
Knoll (based on CGs; see Raphi's comment)
Seth? (Sword in his left hand in a CG; see Raphi's comment)
FE10
Nealuchi? (Holds a cane in his right hand)
Sephiran (Staff user, but brandishing rather than leaning)
Haar? (Leaning on his axe, but that means the right hand would be higher on the axe when picking it up from that position, which would be the hand more in control if he uses both hands.)
Heather
Kieran (See FE9 art; probable since it's a small axe)
Kyza
Nephenee? (two-handed lance? held in left in both FE9 and FE10 art)
Oscar? (lance; left in FE10, right in FE9)
Rhys
Not Sothe (he holds a dagger in his left in his art for both games, but he's wielding it in his right in the RD cutscene)
Titania? (Her left hand is "higher" on the axe in her FE10 art, but the axe is in her right hand in her FE9 art)
Calill? (Holding books in right in FE9, but there are two and it may be heavy enough to warrant the dominant arm)
Despite my headcanon, Pelleas actually looks like a rightie.
FE13
Inigo
Miriel (She is literally holding a quill pen in her left hand, sooo.)
Vaike (axe)
Henry
Ricken
Validar
Small-framed Boy Robin (The in-game portraits are probably meant to be viewed facing-right, per comparison to Chrom's design, level-up default position, etc. That would make this avatar be holding a book in his right hand. This doesn't seem to absolutely hold, however -- Flavia's holding a sword in her right hand in both her official art and her left-facing portrait.)
DLC!Roy
DLC!Micaiah (but she doesn't seem to have joints so who knows if usual handedness rules apply)
DLC!Elincia? (two-handed lance?)
DLC!Eirika? (left hand higher on staff)

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Also, fwiw, in FE6 Roy and Hector both appear to be right-handed, while Marcus is left-handed (based on where he wears his scabbard, which might support the two-handed lance theory). I could buy that Hector's ambidextrous to a certain extent, since he's not consistent in the CGs either.
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The ratios look about right, though, so I think at least most of these characters are true lefties.
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I totally missed Hector, probably because he's holding it in his right hand in his FE6 art, but he is just leaning on it (which might mean nothing). He could also be ambidextrous which would be interesting. Considering the way Lyn yells at him for "swinging his axe around" it wouldn't surprise me.
I hadn't thought of the tomes but that's damn interesting. Ahhh.
I like your random posts like this I need more. They keep me thinking.
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I think this CG backs up your Kyle thoughts? And suggests that Seth can use a sword at least with both hands.
Saleh looks like a rightie here - maybe ambidextrous? Or maybe he's doing something else with his other hand under his cape.
And for the sake of completion since I said it over AIM - Ephraim swaps off with his weapons and gestures, too. Even his sprite trades when he promotes, and the critical makes a show of it.
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Saleh and Ewan both look rightie in that CG, yeah. Well, they didn't have a whole pile of evidence for being leftie, so I'll buy it.
Maybe Seth is pulling the ol' "I'll use my off hand as a handicap" trick. EDIT: Actually, I don't think Seth shows left-handedness anywhere. His CG has his right hand on his lance.
Ephraim is such a show-off. :P
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(... My essay's due in three hours and it's still half undone.)
EDIT: You should totally look and post about it though.
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I'd always headcanoned Knoll as leftie, so this pleases me c:
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There are a lot of mage lefties. :o Soren's a probable rightie though.
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I don't know what all it means, but hey, figured it might help.
Oh, it also shows Tharja specifically casting with her left, so the aforementioned book/hand theories are probably quite accurate!
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