Yes, I am sure that they never used barding (given that I cannot read japanese, but I have never seen any, and have repeatedly heard that they never did.) Also remember how little the the japanese native horses were. If you read any of Friday's books, he goes into this quite well. I don't believe that a japanese horse could actually carry much barding, but in reality they probably didn't use it because no one else had it in japan. Same reason the japanese NEVER used shields. The only shield they ever used were pavise type things. So remember that Japan is japan, and is quite separate from the rest of the world, and asia, in terms of weaponry. No shields, barding, one handed swords, or mail (japanese mail being quite different).
Except for a shaffron, your not going to find any barding, and since early armor used no leg, foot, knee, or dedicated thigh armor (they had large tassets to guard their body in the early years, but no haidate to protect their thighs specifically. So a long slashing weapon would do very well against cavalry. Also you probably noticed that the naginata was also used from horse back.