Halliday & Matthiessen (2014: 547):
Jill says something; this is a verbal event. To represent it, I use a ‘verbal’ clause Jill said, plus a quote of her verbal act ‘It’s raining’. The two have equal status (paratactic), because both are wordings. That is to say, both my locution Jill said and Jill’s locution it’s raining are lexicogrammatical phenomena. Fred thinks something; this is a mental event. To represent it, I use a ‘mental’ clause Fred thought, plus a report of his mental act (that) it had stopped. The two have unequal status (hypotactic), because one is a wording while the other is a meaning. That is to say, my locution Fred thought is a lexicogrammatical phenomenon, but Fred’s idea ‘that it had stopped’ is a semantic one.
Thus parataxis is naturally associated with verbal projections and hypotaxis with mental ones. But, as we have seen, the pattern can be inverted. I can choose to report a verbal act, presenting a locution as a meaning; and I can choose to quote a mental act, presenting an idea as a wording. If we report speech, we do not commit ourselves to ‘the very words’: if I say Henry said he liked your baking, you would not quarrel with this even if you had overheard Henry expressing his views and knew that what he had actually said was That was a beautiful cake.
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To be clear, interdependency status turns on whether or not a clause in a nexus can 'serve on its own' (p509). Here Matthiessen claims, without supporting argument, that interdependency status turns on whether or not clauses in a nexus represent the same level of content, and concludes that there is a natural relation between verbal projection and parataxis and mental projection and hypotaxis. That is, the argument is an instance of the logical fallacy known as petitio principii ('begging the question'), since the argument's premises assume the truth of its conclusion.