This paper explores various policies to support climate-neutral hydrogen production, focusing on their interaction with energy markets and cap-and-trade systems such as the EU emission trading scheme. We develop and deploy a state-of-the-art equilibrium model to examine the effect of hydrogen support policies on the interactions between hydrogen, electricity and emission markets. Our analysis shows that mechanisms remunerating hydrogen production can distort spot prices of electricity and hydrogen more strongly than mechanisms that remunerate hydrogen production capacity. Hydrogen support mechanisms furthermore promote renewable electricity production and deter investment in conventional generation assets. The associated decrease in emissions in the power sector leads to an increase of emissions in the industrial and hydrogen sector due to the waterbed effect in the EU emission trading scheme. Our case study on an emission-capped area inspired by the EU shows that the operational distortions that production-based mechanisms exhibit, typically increase costs more than the investment distortions that capacity-based mechanisms entail.