Unlike Josh, some might say I am hung up on Koryu... and to that end, one of my criteria for an "authentic" Kenjutsu system is one that was born out of experience. As a result, any authentic Kenjutsu system has to be Koryu, or based strongly in Koryu, as there really isn't the opportunity (and, more importantly, the need) to go and create a new system based in experience in using a sword in genuine combat. So that's the light I'm going to look at this in.
There are a large number of things that jump out at me. Firstly, there is no record of any Mukushin Ryu that I, or anyone I've come across, has ever heard of. That's rather unusual for a system that has a history as long as this one claims, especially when it claims connection to well known and established Ryu-ha such as Chujo Ryu.
Next, while the claimed history isn't too "out there", there are articles from organisation, such as this one
http://www.mukushin.com/samurai_dayori_13.pdf which are rather odd in places. The claims which basically go against known aspects of Chujo Ryu (it's Kodachijutsu), instead saying that they are actually their techniques, and Chujo Ryu (quite famous for their Kodachi, by the way) never had any is just weird. Then there are claims such as 'In the first place "Tachiai" means to face your enemy with your sword in your hand" in a section on Iai, which is demonstrably wrong. Tachiai means "standing techniques", literally "standing/upright meeting", and has no connection to the term for a sword "tachi" other than being a synonym. The history then given of Hayashizaki Jinsuke was a little off in a number of ways, and the claim that "His Iai is still alive in every style which has the word Iai in their names until now" is flat out wrong, as there are quite a number of systems that use the term but are unrelated, and the contention that it was developed during the Sengoku Jidai (which is irrelevant to Iai's development, as it's not a battlefield art, except in the case of the Shojitsu Kenri Kataichi Ryu), when Kenjutsu was "undeveloped and neglected" could tick off memebers of, say, Tenshinsho Den Katori Shinto Ryu, to name one...
Finally, the demonstrations that I've seen, all the way up to the head of the system, are universally sub-par. The current (only?) head demonstrates some Tameshigiri against bamboo on their webpage, with badly angled and overly muscled cuts, showing a rather large lack in his ability. The techniques themselves are quite odd in a number of ways, with extraneous and unnecessary actions (turning the blade edge up while still in range of the opponents sword at the end of the technique before moving out and performing a form of Chiburi, for instance, or the out of range attacks and counters which serve no purpose and show large gaps in understanding of distance, timing, hyoshi, and more, ending with the simply poor technique (cutting angle, use of hips and tanden, hasuji, and more) shown by everyone involved leave me feeling that there's a lot lacking here.
What all of this means is that, while they aren't anywhere near as obvious as a number of other made-up systems, there's not a lot of supporting evidence for any actual authenticity with this group. Treat with caution if you approach them.
EDIT Out of interest, what happened with the Yagyu Shinkage Ryu group I linked for you last time? That's a guaranteed legit system for you right there!
Mike Femal has a Toyama Ryu dojo in St. Petersburg ... http://www.toyamaryu.org/
