I had written the same thing, then wussed out after failing to find it in about a dozen long lists of phrasal verbs. Curiously,
it was used in the definitions of many other phrasal verbs.
It has a verb and an uninflected adverbial particle, if you consider 'started' to be adverbial in the phrase, so I suppose it qualifies. I'm not comfortable with the adverbial characteristic of 'started'.
Surely the 'get' is modifying the 'started'? You could say 'I will start' or 'I will get started' Therefore the 'started' would be a past participle and the 'get' an adverb. Mind you, in my sentences it's future tense. Perhaps 'get' is an auxilliary? Oh I don't know, it's complicated!
Complicated indeed, but it is not possible for "get" to work as an adverb. Verb forms not working as verbs can only work as nouns or adjectives: gerunds or participles. Adverbial useage need not apply.