Recycling the revenues of a carbon tax can mitigate the distributional impacts and lowers the burden on the lowest income deciles. However, a lump-sum rebate to households induces consumption, hence emissions. In this paper, we study the existence of a backfire effect where emissions increase above the pre-tax level because of the recycling of carbon tax revenues. We build a small theoretical model that we extend using microsimulation on French Households Budgets Surveys with long-term elasticities. We estimate that a €158/tCO2 tax would induce a decrease of 10.9% in emissions, reduced by a uniform lump-sum rebate to a decrease of 5.9% in aggregate emissions. We conclude that the backfire effect is not a sufficient reason to prevent any compensation of the low-income households for the sake of emissions reduction. Indeed, a quarter of households increase their emissions in the face of a lump-sum rebated carbon tax, but the emissions thus emitted represent less than a tenth of the emissions of the 10th (richest) income decile. Recycling only 60% of the carbon tax revenues would reduce the backfire to decrease emissions by almost 8%. Skewing the recycling towards low-income households increases the progressivity of the tax but does not increase emissions. Our study supports the consistency between reducing inequalities and reducing emissions.