I believe I've found a decent video of the Kumi Tachi that Richard has been demonstrating.   Saotome Kumi Tachi #.  Some notes:
  1. Prior to the strike, the sword drops down to "touch the spine" - I really don't understand this, and I think it is time to consult Flemming Sensei. I'll just have to organize my thoughts so I know what question I'm asking.
  2. The video is shot from an angle, which means we can't see the movements back and forth off the line of attack, and we can't see the contention of the two participants for the centerline.  That is a great pity.
  3. I love the fluidity of this technique; the second iteration in particular flows from one technique to the next. Wow. Just wow.  Difficult to imagine the path from where I am now to that kind of flow.  (Let me pause and consider what Richard feels like dropping back to our level.)

Some potentially good videos for Kumitachi
  1. Tiny picture of the Kumitachi - this site has them broken out to study in isolation, but I don't think the size of the picture will help me to learn.
  2. Kumi-tachi 1-5 - Looking only at #1, this doesn't begin the same as we've been practicing - the first cut is a simple shomen; we've been starting with a cut that looks kind of like aiki-ken #3.
  3. Saito Kumitachi - I think all the same movements are there, but I'm having trouble following. In particular I simply can't see the subtlety of the final cut.
  4. TAKN de Quiros Sensei is one of Saito Sensei's students, and the relationship is obvious.  Many of the same habits that I find confusing from Saito Sesei.  (E.g. the vertical sword prior to cut, and the horizontal finish).  Once again the exercise begins with a simple shomen cut rather than the waki cut.
Sadly, it appears that MuGai Ryu has a completely distinct set of kumitachi  Example #1 and Example #2

We've been studying Saito-Sensei's aki-ken suburi.  We'll be ready to move on to the Kumi-tachi  soon. Mr. Gejji has been demonstrating as kumi-tachi #1, and I've found it a really good exercise for seizing the center and for learning about distance.  I will admit to some confusion because I don't think it looks like Saito Sensei Kumi-Tachi #1I think it is possible that his version of Kumi-tachi matches  Saotome's Kumi-Tachi #1.
Something to research