Jodo, on the other hand, is never done on the other hand. The most basic explanation for this is the old adage, "There is no left hand in Japan." The story goes that feudal Japan was such a homogenous society that left-handedness was repressed. Children were simply not allowed to grow up left-handed. So there were no left handed swordsmen.
Now that story might be apocryphal, but what is apparent is that jojutsu masters did not waste time trying to make the system symmetric. The kihon exhibit several different kinds of sidedness just like the judo throws described previously, but the seitei kata, like the one demonstrated below, are only ever done one-sided.
Occasionally for a mental stretch or as a cool-technique-of-the-day, I'll flip one of the kata to the other side and rep it a few times. But that is not really an attempt to bring my off-side up to speed but rather just a mental exercise. I've found that once you practice enough of the kihon on both sides then you can flip a kata on the fly pretty easily.