A brief update to this post from 2016.
Several Japanese news outlets reported today that one of the swords that were discovered during 1939 repair measures of the Kasuga-taisha turned out to be a Ko-Hōki (古伯耆) from the end of the Heian period, possibly a Yasutsuna (安綱), dating back to the birthplace of the nihontō. The blade was polished recently, is ubu I think, and has a nagasa of 82.4 cm. Its kokushitsu-tachi koshirae (黒漆太刀拵) dates to the 14th century and it is assumed that the sword was offered to the shrine some time in the Nanbokuchō or at the latest at the beginning of the Muromachi period. Below are some links with pictures and please also check out the video that the Nara Television Channel put on YouTube.
https://mainichi.jp/articles/20180123/k00/00m/040/109000c
https://www.nikkei.com/article/DGXMZO25985170S8A120C1CR8000/
http://www.mbs.jp/news/kansai/20180122/00000058.shtml
Markus Sesko 
It is most likely that these swords were hidden by the shrine to escape confiscation from the US occupation authorities, hence them being placed were they were until recently and overlooked. One of these (like this Ko Hoki) was designated Juyo Bijitsuhin in Showa 14 (1939) but for some reason were not repolished then.
Holy smokes, this is the news i do like to read. Is this one of the (if i recall correctly), 4 blades that were in the ceiling somewhere?
Indeed, another great discovery. By taking a look at the blade, the possibility of Yatsutsuna is rather low. Instead, this is more likely to be a work of Nichijo (日乗), Aritsuna (有綱), or Sanekage (真景).