Speaking for the Greeks - we don't know. The concept of left-handedness in combat is never addressed by the surviving sources. However, the fact that most people are right-handed (and therefore carry their shield on their left arm) plays an important part in how the phalanx works; I wrote more about this here. Given the vulnerability of a hoplite formation to attacks from its right (unshielded) side, we'd expect that the creative use of a group of lefties would show up in the sources. Since there is no evidence for anything like that, it's fair to assume that Greek armies did not make any deliberate use of the fact that not everyone fights best with their right hand.

We may assume that there were cultural reasons for this lack of acknowledgement or use of left-handedness. Modern authors have noted that the Greeks maintained a very clear dichotomy in their moral and philosophical thinking, in which all things "right" were seen as good, strong, healthy, manly, and auspicious, while all things "left" were weak, sickly, bad, feminine, and ominous.^1 It would not be an overstatement to say that the Greeks lived in superstitious fear of the left side of things, to the point where they even tried to avoid saying the word. In literature, the left is sometimes referred to as euônymos ("the honoured side") or aristeros ("the best side"), words which are to be understood as euphemisms meant to ward off evil and bad luck.

It's not clear to what extent these prejudices would have affected the Greeks in their practical decision-making. For one thing, contrary to the argument of Lévêcque and Vidal-Naquet, Greek generals seem to have had no issues with deploying their best troops on the left flank. For another, Greek architecture and town planning is famous and influential specifically because of its symmetry; whatever moral prejudices the Greeks may have had, their ideal of beauty and order was actually predicated on the equality of left and right. It's difficult to know for sure, then, that their view of the left as sinister (an English word derived from the Latin for "left") would have made them hesitant to identify or use left-handed men in wartime.

However, if we suppose that prejudice against the left was above all personal, rather than social or professional - that it affected the way people saw themselves in the world, rather than the way they tried to affect that world - it may well be that left-handedness was not an issue for adult male hoplite levies because any such tendency would have been surpressed in childhood. This was the reality in many Western countries in more recent times; those who showed a natural inclination to write with their left hand were forced to learn to write with their right. It's possible that something similar was the reality for all Greeks growing up, and that there simply were no left-handed men - only those who were good at being right-handed and those who weren't. It's also possible that left-handed men would be so ashamed to show their supposed disability in public (and to raise suspicions as to their moral character) that they would do everything they could to hide it.

This is of course all speculation; as I said at the beginning, there is simply no evidence of left-handed hoplites except by artistic convention/error. But perhaps the above can explain the curious lack of attention Greek authors paid to this subject, despite its relevance to their military practice.

1 - P. Lévêcque and P. Vidal-Naquet, ‘Épaminondas Pythagoricien ou le problème tactique de la droite et de la gauche’, Historia 9 (1960), 294-308