It is important to separate the two kinds of intuitions. Suppose you are initially inclined to intuit thus: "the property of being a dog is intrinsic". This might well be because you are simply presupposing the worm view [of 4-d objects], and thus presupposing that 'being a dog' refers to the property of being a dog-worm, and metaphysically intuiting that this property is intrinsic. It might well be that, once the question is raised of whether the worm-view or the stage-view is correct, your initial conviction disappears, and is replaced with agnosticism about whether "being a dog is intrinsic", despite your certainty that being a dog-worm is intrinsic and being a dog-stage is extrinsic.
[From his paper on counterpart theory: 'Beyond the Humphrey Objection'.]