If you check those properties (set them to true in ASPX templates) it may seems that links to previous/next and first/last items don’t appear on the page even though you would expect them to show up in the pager.
Actually it could be correct behavior, however one must admit it's a little bit confusing. Documentation says this: If checked, first and last buttons of the pager will be displayed only where usable. When you set DisplayFirstLastAutomatically to true it means that first/last links will be displayed only if there is no other possibility to reach the (first/last) page. It could be in the situations when you have set e.g. GroupSize to particular number and there is still page in another group of results.
The same applies to DisplayPreviousNextAutomatically likewise.