Paracelsus, I promise that my flames will be accompanied at least by some good sauce
Jim, according to a web search:
http://www.uh.edu/admin/engines/epi476.htm
stirrups were known to the Chinese at the time of Christ and to the Europeans (Charles Martel) around 732 AD, so I'd have to think that the Japanese samurai had them in the 13th century.
So it sounds like the horse mounted soldiers in your fictitious empire would probably have an edge over the foot soldier (be they Spartan or Roman trained). On foot may still be a different story tho.
I'll defer to you on the availability and quality of the steel. I don't recall seeing a lot of drawings or descriptions of steelwork for ancient China or Japan. That may put a damper on your Spartan/Roman shields too.
And no, I don't think there were any crossbows around, nor of course the longbow, but the Asians certainly did have a bow available, probably before the stirrup, tho this is not the issue, since you've already made the point that the Spartans could defend themselves against the arrow using their shields (if you have enough metal to make them).
Your mention of the use of the katana in close quarters was interesting, and I wonder if it has any bearing on the discussion, particularly in wooded battles and melees.
I don't have a problem with the idea of a phalanx being effective, I just wondered how often the situation would lend itself to allowing its formation.
And the dope on Hastings was also interesting, but in our case, remember that your hypothetical adversaries are equally competent and rested as yours. All that differs are the weapons and the methods of their feudal warlord master ... you.