Reijyu or Spirit Beasts (sometime referred to as Riding Beasts) are magical creatures that have special powers (be they as simple as being able to run fast, or as complext as the ability to consume energy from attacks). Here's a list of Reijyus that are found in Houshin Engi:
Shuupuushan was initially given to Taikoubou as his own spirit beast. Shuupuu's a very deligent and loyal reijyu, who always stood by Taikoubou through all the chaos and idiocy. Basically, Shuupuu is the sensible one of the two. Shuupuushan comes from a family of 3 (him behing the youngest and only son). The Shuupuushan family has loyaly followed Geishitenson's command for generations, hence Shuupuu feels it is his honour in serving Taikoubou.
Shuupuushan has the ability to transform into his adult form (as shown in the picture above), which is much bigger in size and more powerful in strength. In his normal form, he can't really do much except fly. But in his adult form, he has the ability to absorb (or should I say "EAT") pure sources of energy. For example, he *literally* ate the rainbow dragons of Chou Kou Mei's super paopei, and he also freed Taikoubou/Fukki by eating Jyoka's paopei.
The above picture depicts Shuupuushan's parents (Shuupuushan Papa and Shuupuushan Mama) in their normal forms.
Shuupuushan Papa was the Reijyu of Geishitenson. Supposedly, some hundreds of years ago, Shuupuushan Papa was instrumental in the Geishitenson's victory against Chou Kou Mei during a Sennin fight that occured. Unfortunately, that same fight left Shuupuushan Papa heavily injured and he has never truly recovered from it. Hence he doesn't fight anymore.
Kokukirin is Bunchuu's faithful Reijyu. Kokukirin often act as a messenger for Bunchuu, such as in the example of the ShiSeis. other special abilities of Kokukirin is its incredibly tough exo-skeleton. It was able to protect Bunchuu from all the blasts from the Kunrun forces in the Sennin Kai battles. However, even Kokukirin has its limits. It died trying to protect Bunchuu from the acid rain drops of Ou Tenkun's Kousaigin paopei.
Known as the "Ultimate" Reijyu that ever lived, Kokutenko has the ability to see far away into the distance (his "Senrigan" or Thousand Mile Vision) and hear things millions of miles away. He is also incredibly fast and quite intelligent, however, he is still unable to decipher his master, Sinkorhyo. Other than that, we never get to see what sort of special powers Kokutenko have. Just like his master, Kokutenko is just as secretive and mysterious, though much cuter. ^_^
Koutenken isn't exactly a reijyu, rather, it is more of a paopei, due to its destructive abilities. However, when ever Koutenkun is not called upon to destroy something, Youzen often uses it as a riding beast. Don't be fooled by its fluffy, cute expression, Kouteken packs quite a punch when need be. It can smash through almost anything.
This is Uen, the riding beast and reijyu of Chou Kei and Kouranei. Uen's only special ability is to run really fast. It leaves massive foot prints where ever it has walked through. Because of its speed, it could probably walk on water (for example, Buchiki ran on water as well), which is probably how it reached Dakki's floating shenkai.
Shinyu is a reijyu (Spirit Beast) but not exactly for the purpose of riding on. Due to its small size, it cannot be used as a riding beast to carry Suukokuko around, however, this does not mean that it is no longer considered as a "reijyuu" (which literary means "Spirit animal"). Shinyu is able to command all the species of birds into doing what ever it wants them to do. It is said that Shinyu is more well known than Suukokuko. ^_^;;
Technically, Ryuushuko is NOT a spirit beast. Instead, it (he or she, not sure) is often called a magic beast, or a magical creature. I don't really know what's the difference between the two, but it is noted in the story that Ryuushuko is not categorised as a 'spirt' beast, instead he refers to him/her-self as a magical beast. Ryuushuko is the best friend of Tou Sengyoku (a female Sennin whom trained in the Kingoto Islands for about a decade, she then returned home and worked under Bunchuu as a spy gathering information on Taikoubou and co.). When Sengyoku was called back to the In to work under Bunchuu, Ryuushuko went with her. And when she left on her spying mission, Ryuushuko stayed with Sengyoku's father - Tou Kyuunou (hence the pictures where Tou Kyuunou standing on Ryuushuko's head). We don't really know much about Ryuushuko other than the fact that he/she is very protective of Sengyoku, and can use his tail to flip huge bolders at an enemy. Definitely not an easy foe to overcome.
If there are any more questions you have about anything relating to Houshin Engi, please email me your question. I can't promise that I can answer everything but I'll do my best. ^_^