Scu pointed me to this discussion of a Cap and Trade dividend.
One issue, which would also apply to a carbon tax + dividend: this would hit exports but miss imports. Contrast that with a plain-old VAT, which hits imports but is refunded on export. This would make our goods less competitive, while encouraging us to import goods that are carbon intensive, externalizing our environmental effects. Of course, we’ve already been doing this: that’s why China is a larger carbon-emitter than the US: they emit our carbon for us. But this would exacerbate that effect.