We upgraded directly from 5.4 to 5.6 with an in place upgrade using the "-unpack" method. We did the Author on a Saturday, and the Publishers on the Sunday, mostly because we ran into the Publisher permissions problem on Saturday night, and stayed on 5.4 publishers while trying to resolve.
Anonymous User Permissions on Publishers are more restrictive, pretty much just has access to /content and /etc on the publishers in 5.6. So any assets that may have been used from /apps even if not through the public interface should probably move under etc. We had XSL transformations that the component couldn't read the .xsl file under /apps. I think this also is the root cause of the ResourceResolver issues.
