I'd always blithely assumed drawn here meant 'pulled' - as used in the context of wire-making, where malleable metal is 'drawn' through the die/plate/orifice to elongate into lengths of wire.Probably this understanding came to me from drawings of four horses drawing/quartering the victim.This still makes better sense to me of the sequence- with drawn preceding quartered, and there is some debate about the exact meaning. But looking it up now on more than one website, it would seem that it is, by and large, accepted that the word 'drawn' is being used here in the more familiar sense of being 'drawn' / carried along by horse/s - to the scene of the execution. IMHO, Dragged would have been a better choice for such a wretched context!
It seems illogical on the face of it, but the word hung can come first in the sequence, when you consider that the condemned was not hanged 'till dead', but lowered to the ground in a half-dead/semi-conscious stage.
In the denouement, his limbs were fastened to four horses that were whipped hard to send them scampering at different angles, to effectively 'quarter' the subject, aided by some strikes of the executioners' swords to help sever the limbs. It seems disembowelment was an optional extra- carried out either at the discretion of the executioners, or by specific imperial diktat.
'Drawn and quartered', is not always preceded by hung, AFAIK. Maybe the verdict depended on the severity of the crime / wrath of the king. It seems plausible that a traitor could either be just drawn and quartered, or, to really make a spectacle out of it, first hanged, then 'drawn' and quartered. And sometimes eviscerated as well.These were usually public executions, witnessed by scores of onlookers. For those who couldn't make it to the show, the severed limbs were left dangling at the venue. Sometimes the heart and limbs of the condemned were taken out in a procession and displayed to the public.
This one has a somewhat graphic description: