Halliday & Matthiessen (2014: 184):
We can recognise two types of modal Adjuncts, (i) mood Adjuncts and (ii) comment Adjuncts. (i) Mood Adjuncts serve within the Mood element, and are closely associated with the meaning of the Finite element – the limiting case being modality, which (as we have seen) can also be realised by the operator serving as Finite. (ii) Comment Adjuncts serve outside the Mood + Residue structure of the clause. They are not part of the proposition realised by Mood + Residue, but are instead comments on it (propositional) or on the act of exchanging it (speech-functional). These different types of modal Adjuncts are characterised by different grammatical properties, including different agnation patterns in terms of possible alternative forms of realisation…