Yes, it is a bit of a challenge to answer that one.
There is no question that early Japanese swords have links to Chinese forging and style methods, roughly around 300 BC to 300 AD (And after in varying amounts and areas)
The two countries actually enjoyed many years of mutual collaboration and cultural exchange before the many wars that created so much bad blood between the two.
I should note that the Katana we know today is the result of many changes to the initial Chinese style Japanese swords of the ancient eras through uniquely Japanese influences..
It is like how early Chinese martial arts have been traced back to origins in india, but the styles of Kung Fu we know today are uniquely Chinese if you catch my drift.
Now on to the sword.
First, is this the style you are talking about?
http://www.tnm.jp/uploads/r_collection/LL_C0035224.jpg
This is a typical ornate Katana from the Heian period, note the Chinese style guard
This style was of sword was often called a Kazari-tachi, Kazari refers to something of an elegant style and Tachi to the type of sword.
All swords of this era are also called Koto, which is more of a modern age reference, with different names applied to different eras
Now these type of swords were mostly worn by nobles in ceremonial situations such as in courts and lavish affairs, more as a sign of status and wealth than as a practical weapon.
Because of that status, there is no question of its use as a gift or heirloom which makes it perfect as a religious artifact.
(As nearly all artifacts are gifts or heirlooms from great or respected people from various eras)
However the Tachi types were still used in battle, often worn by officers on horse back and cavalry units, and by that time were a less ornate type with more of a Japanese feel called a Hoso-Tachi or Efu-tachi. (Hoso being quite thin and Efu having the more Katana like shape and disc style Tsuba)
The blade was worn sharp edge facing down to facilitate easier drawing from a seated position in the saddle, but was more of a secondary weapon used when the spear was lost or broken during battle.
Earlier versions with a straight blade can be called Chinese style Tachi, but It is hard to arrive at an exact name used in those periods due to the methods and details of ancient records which leave much to theory as well as dealing with varying era and regional dialects.
A good example is the imperial treasure "Kogarasu Maru", a double edged Tachi often called called, "Kissaki Moro Ha Zukuri" in relation to the blades shape rather than the exact name of the swords style itself
Naming the type of guard puts us in an even harder position.
While we associate the word Tsuba with Japanese Katana now, its base meaning is simply referring to a swords guard.
So technically we can call it a Tsuba regardless, and finding the exact name for that style of Tsuba can be tough due to the reasons mentioned above
It is like how Katana is instantly recognizable as the Japanese style sword, even though it's base meaning is reference to the word sword. More technical names are still used but not as strictly these days.
As for finding yourself one of these older style swords, you may have your work cut out for you since the styles are lost among the ages as well as the recent popularity of the Japanese swords we know today. I have seen plenty of rubbish display replicas which don't quite come close, or authentic ones usually in a museum or royal collection.
I can find plenty of Tachi, just having trouble finding ones with the more Chinese style Tsuba (For sale anyway)
Hopefully I gave you at least enough reference material and names to help with your research, good luck.
(Oh and thank you, it's not often I get to touch on real swords and their history here among all the, "Where I get good Katana for $50", kind of questions)