Hi Daniel, thanks for posting with some very clear pictures. I'll answer a few basic questions and I hope someone with more experience in dendrobium culture can give you more advise on its care.
Firstly it is a hard-cane dendrobium, it grows a bit warmer and more humid than the soft-cane dendrobiums. And indeed the swollen vertical 'stems' with pairs of leaves growing on either sides are called 'canes'.
There seems to be a lot of rotten roots on the old canes, but the good news is that there are some very healthy roots emerging from the new growths.
I believe that you have been overwatering it, and the mix it came from seems to hold a lot of moisture. I suggest you trim some of the old decayed roots, and pot the orchid in a medium grade orchid mix in a small pot and keep it on the dry side with just some occasional misting to encourage further root development. The size of the pot should be just big enough to hold the roots and have a distance of one inch from the newest canes to the edge of the pot (Dendrobiums like most orchids do not like to be over-potted). You might need a stake to hold the plant in place as the canes should not be buried, the base of the cane should sit just above the orchid mixture in the pot.
How much light does it get?