A basic requirement for on-off intermittency to occur is that the system possesses an invariant subspace. We address how on-off intermittency manifests itself when a perturbation destroys the invariant subspace. In particular, we distinguish between situations where the threshold for measuring the on-off intermittency in numerical or physical experiments is much larger than or is comparable to the size of the perturbation. Our principal result is that as the perturbation parameter increases from zero, a metamorphosis in on-off intermittency occurs in the sense that scaling laws associated with physically measurable quantities change abruptly. A geometric analysis, a random-walk model, and numerical trials support the result. We will show how this result occurs in many chaotic systems that exhibit chaotic rotations.