I've now read the relevant section. I indeed find the quote on my page 11. The quote appears in Section 1.5, which presents the "Theoretical Framework" employed in the book. This framework is a highly idealized version of syntactic structure that builds on some very specific presuppositions about the nature of syntax.
John Lawler's comment is on the money in my view. One has to swallow a lot to go with this sort of understanding of syntax. In particular, the following presuppositions are present:
All syntactic structure is binary branching.
All syntactic structure is right branching.
Numerous empty elements are present in the structure, i.e. theta heads.
All arguments merge as specifiers, whereby the predicates to which they belong appear originally to their right as complements of the empty theta heads.
Leftward movement (actually copying) occurs on a massive scale in order to achieve surface word order. For instance, verbal predicates must move leftward in order to precede their object arguments, e.g. one wants eat meat instead of meat eat, one wants believe that it works instead of that it works believe, one wants say it instead of it say. etc.
In my view, this is junk science. It sounds impressive, but it's nonsense. In order to buy into this view of syntactic structure, one has to take the leap of syntactic faith that I associate with much of Chomskyan syntax. I even think many Chomskyans would reject the understanding of syntax that is put forward in the book. The approach is, for example, contrary to basic notions of X-bar theory, in which complements (as well as many specifiers) are arguments. On the approach in the book, all complements are actually predicates.
But to answer the actual question, my understanding of the final sentence in the passage is that the head-complement dependency associated with standard X-bar-type structures is actually present, but its presence is manifest as the dependency between the theta head, which is always an empty element!, and the predicate, which is represented by X in the complement position. The empty theta head θ takes a predicate X as its complement; a dependency exists between the two. But since the dependency is unlike the dependencies that are traditionally assumed, Rowlett calls it a "grammatical dependency", a nebulous term.
Hence Rowlett appears to be stipulating two basic types of dependencies:
Grammatical dependencies between empty theta heads θ and their predicate Xs (head-complement dependencies) and
Thematic dependencies between empty theta heads θ and their specificiers (specifier-head dependencies), these specifiers being what one traditionally understands to be the arguments of the predicate Xs.
Rowlett is thus pursuing an approach to syntax that takes specifiers to be the arguments of complements in a sense, as opposed to the arguments of head predicates, which is much more standard. I think such an approach is going to be difficult for many a syntactician to swallow.