The main thing in working with a graph (meaning, digraph, since the distinction is important in this application) is that its loops may be isolated, connected in one direction but not the other, or mutually connected.
With respect to a de Bruijn diagramde Bruijn diagram, the first alternative would mean that there is a simple pattern, persistent or shifting as the case may be, but essentially unique, not admitting any variation. For example, superluminal patterns generally have this form since causality is not operating.
The unilateral connections correspond to fuses, which is an irreversible change of pattern which may be either static or shifting. Many configurations for Rule 110 have this form, including most of the ``speed zero'' shifts, and in particular the C gliders, which can abut on uniform quiescence, or vacuum. But shift 2 in five generations has a different kind of split field --- T1's on the left giving way to a mixture of T3's and T1's on the right, along a right-moving interface. And of course, we have already seen an an almost identical combination in Figure 9.
Gliders depend on there being a loop generating the ``ether'' which has another connection to itself constituting the ``glider.'' The ether loop could be an autolink to the quiescent state, but things are different in Rule 110. There might possibly be several handles, signifying distinct forms of gliders or different phases in he evolution of a single glider, all moving at the same velocity.
You can't tell the ether from the glider without a program; this is evident when looking at the A gliders, for example. T1's can intersperse T3's to get the A gliders, but lots of T1's can harbor an occasional T3 for a role reversal. Of course, the T3's figure in lots of other configurations, so it is reasonable to assign them to the ether.
A still more complicated combination has the glider off in a loop of its own, but still having mutual connections to the ether loop. That is the arrangement with respect to the extensible gliders, and can be used to determine admissable spacings, closest approaches, and so on. And it is a property of regular expressions, that if the glider is once extensible, it is multiply extensible.
Once someone is familiar with these ideas, it doesn't really require constructing de Bruijn diagrams to take advantage of the information; however it should make it less surprising when these relationships are observed in practice.