I would like a PDF reader that lets me split the screen so I can see two different parts of the same document. I do not want a split view that only lets me see two consecutive pages.
This is a bit of a strange answer, but Konqueror (the default everything-browser for KDE) can do what you want. You see, Konqueror is not just a web browser or just a file browser -- it is a browser for any type of data that there exists a KPart for (where a KPart is some type of component that KDE can understand and embed places -- things like Konsoles, Kate windows, and -- of interest on this occasion -- KPDF viewers).
Thus, because Konqueror has a split view, you automatically get a split-viewable PDF reader for free. Just open up a Konqueror window, split its view, and open up the PDF in each pane. Each can scroll independently of each other, etc.
You'll have some of the basic PDF-browsing features but not some others -- KParts are not meant to be full applications. Thus, you won't (for example) be able to annotate the PDF in Konqueror as you would be able to in Okular, for instance. But as long as you just want basic viewing, this will be enough :)
KParts and all the modularity of the KDE framework allows for some pretty neat stuff, in my opinion.
Emacs 23 (and earlier versions with the separately obtained doc-view.el package) can seamlessly open and view PDF files. And using the usual split window support (C-x 2) you can display two different parts of the same document in the same window.
I doubt that this is going to be as easy to use as just opening a normal PDF twice, but it definitely does exactly what you ask for.
If you want to split the pdf in half and see each half in a screen, you can try dspdfviewer.
This is a kind of hack, but you could run LibreOffice Calc and use Insert>Image... to insert the document. Then use the Calc's splitting feature by dragging the small dark rectangle at the right top or right bottom area of the sheet.