Since this is an informal piece, I am not going to be anal about citation. Know that I am making explicit reference to materials, which I name in the text. If you want page numbers or specific scenes, find them yourself.
That The Legend of Zelda is twenty-five years old seems somehow to have eluded me until a day or two ago. As it came to my attention, I thought back over some things that I remember from having played earlier games in the series--the original, Link's Awakening, Link to the Past, and Ocarina of Time--and it occurred to me that Link, for the most part, wields his sword in his left hand.
I do not know if anyone else has commented on it, and I am not particularly minded to care at the moment (although if anyone has, I should like to be informed), but it does seem to me that there are a few things it can mean. Perhaps it is a commentary on the evils of physical violence. The English word "sinister" derives from the Latin word for "left" (that is "left" as opposed to "right," rather than "left" as in "behind"). That the sword is put in the left hand can possibly be an indication that physical violence is, well, sinister, and thus to be avoided. That so much of the Zelda series relies on solving puzzles rather than simply smashing heads--and even the head-smashing tends to require some thought to do correctly--and the perceived dichotomy between the violent and the cognitive* suggests that this might be so.
It might also be nothing more than a reinforcement of the surreality of the milieu in which the games take place. Most people are right-handed, and so tend to take up arms in their right hands. The default setting for items, as southpaws can attest, is to be used right-handed. The world, really, is set up for righties (except baseball, which plays both ways). That the worlds of the Zelda games are set up for a lefty deviates from the expected norm of the "real" world, giving an indication that the game is a "fake" one.
As though 8-bit glory and its many children need delineation as fiction.
Since the right hand, in the Japan that gave rise to Nintendo and The Legend of Zelda as in Western culture, is privileged and the left hand is disfavored, there is, no doubt, significance in the assignment of shield to right hand and sword to left. Perhaps it is a commentary on relative valuation of defense and offense--putting the shield in the right hand would tend to indicate that it is preferable to defend than to attack. Certainly, there is some traditional Western precedent; in Malory's Le Morte d'Arthur, for instance, Merlin comments to Arthur that the scabbard which accompanies Excalibur is more valuable than the sword itself. As it permits the bearer to sustain injury without suffering harm, rather than allowing for inflicting harm, its valuation above the sword indicates that effective defense is preferable to effective offense.
Of course, it could also be nothing more than a programming quirk. But that would not be nearly as interesting.
*Of course, thought can be violent, and it is not necessarily true that mind and body are so wholly dichotomous as is sometimes thought.