I have an svg in which I was able to change all of the nodes to either symmetric or smooth; however the two indicated by the marker will not change to smooth or symmetric. Instead of having just two handles I end up with four handles when I try to convert to them to either smooth or symmetric. What kind of nodes are these and how can I change them?

They aren't single nodes - they're pairs of nodes. You can see this by selecting them (drag a box around them) and looking at the node count on the status line.

So select one pair, and click the split button. Then select them again, and click the button to join them as one.

Then do the same for the other pair.

sas I appreciate that very much. I knew something was wrong, but I was only trying to just join them and inkscape would let me know that I could not do that. You learn something new everyday.