… 'natural' is not a way of avoiding work, it is a way of defining (often quite hard) tasks. The 'natural' relationship referred to is that there are structural and functional similarities of a revealing and useful kind between the apparently distinct domains. Semiotically we are mostly situated here in Peircean metaphor, i.e., the third and most complex kind of iconic relationship, which is always something constructed … rather than simply present. And these can go in many different directions, yes.
 To be clear, the word 'natural' is not a way of "defining tasks" — "quite hard" or otherwise. In its use with regard to the relation between semantics and lexicogrammar, the closest of its non-technical meanings is 'entirely to be expected'; see further below.
 To be clear, in SFL theory, the relation between semantics and lexicogrammar is 'natural' in the sense of being non-arbitrary. Halliday & Matthiessen (1999: 3-4):
A systemic grammar is one of the class of functional grammars, which means (among other things) that it is semantically motivated, or "natural". In contradistinction to formal grammars, which are autonomous, and therefore semantically arbitrary, in a systemic grammar every category (and "category" is used here in the general sense of an organising theoretical concept, not in the narrower sense of 'class' as in formal grammars) is based on meaning: it has a semantic as well as a formal, lexicogrammatical reactance. … Grammar and semantics are the two strata or levels of content in the three-level systemic theory of language, and they are related in a natural, non-arbitrary way.
 To be clear, Peirce's notion of 'icon' is irrelevant to the relation of semantics to lexicogrammar in SFL theory. Not only is Peirce's theory of semiotics based on assumptions incompatible with SFL theory, the iconic relation obtains between content and expression, not between two levels of content. Moreover, Peirce's metaphor is, strictly speaking, a type of hypoicon:
An icon (also called likeness and semblance) is a sign that denotes its object by virtue of a quality which is shared by them but which the icon has irrespectively of the object. The icon (for instance, a portrait or a diagram) resembles or imitates its object. … Peirce called an icon apart from a label, legend, or other index attached to it, a "hypoicon", and divided the hypoicon into three classes: (a) the image, which depends on a simple quality; (b) the diagram, whose internal relations, mainly dyadic or so taken, represent by analogy the relations in something; and (c) the metaphor, which represents the representative character of a sign by representing a parallelism in something else.