I've seen a little bit of sniping about whether it's Basidiomycetes or Ascomycetes that "really" have a hymenium. There is some precedent on both sides. After all, the Basidiomycetes start off (but for how long?) with the class Hymenomycetes, so it would be rather silly if they didn't actually have a "real" hymenium. On the other hand, the class Hymenomycetes was declared before LÚveillÚ, that is, at a time when all fungi were thought to be Ascomycetes.
In any case, when it's the Ascomycetes that are supposed to not have a real hymenium, they have a thecium instead; and when it's the Basidiomycetes, they have a pseudohymenium. I wonder what these people would call the layer of conidiophores lining a pycnidium. Anyway, few people bother with this argument anymore, as few people work with both phyla. Everyone just calls the fertile surface on their own fungi a hymenium, and doesn't care what the other researchers call theirs. It may seem obnoxious of me to dredge up this obsolete controversy when no one really cares about any more; but you may run across it in the older literature, and I believe that it is the source of the similar, stupid, and continuing controversy over names for hymenial cells.
If (as in this picture) the non-basidium cells making up the palisade look like smaller versions of the basidia, they are called basidioles; the rationale is that they may be immature basidia. Otherwise, they are called paraphyses.
There are those (still) who insist that the cells in a Basidiomycete hymenium can only be called basidioles or pseudoparaphyses, and the word paraphysis is only properly applied to the same sort of "filler" cell in an Ascomycete hymenium. Why, I have no idea. After all, there are many Basidiomycete hymenia where most of the cells obviously have nothing to do with the basidia, although (as here) cells that are clearly immature basidia may be present. It is silly to have to call the long branching hyphae in this picture basidioles (or invent a new term for them) just because some people want to reserve the term paraphysis for use by ascomycetologists. There are also other problems with the use of the term pseudoparaphysis in this context, which will be discussed below.
I should mention here that some Holobasidiomycetes have giant, distinctive cells protruding from their hymenium. These cells are called cystidia (or setae), and have their own entry. See this entry also for clarification of the Holobasidiomycetes.
The cells in an Ascomycete hymenium tend to be longer and thinner than those in a Basidiomycete hymenium. The ones that are not actual asci are called paraphyses.
If you don't have a definite need to know, you're going to want to stop now, take my word for it. Otherwise, you'll just get confused and frustrated.
Ascomycete paraphyses are often long enough to resemble actual hairs. Cells of this nature that occur, not actually among the asci, but towards the edge of the hymenium, at the mouth (ostiole) of a perithecium containing the hymenium, are not called paraphyses, but periphyses. Also, sometimes long hair-like cells grow down from the roof of locules (but not perithecia), and often end up connecting the roof and the floor of the locule. These cells are called pseudoparaphyses, and they are the other reason that calling Basidiomycete hymenial cells paraphyses is a bad idea.