Conservatives

What should hermaphrodites do?

Nothing of the science requires any social distinction on the basis of what gametes (if any) an individual produces.

Any such distinction you - or anyone - cares to make about any individual is not based on science.

Biology isn't even relevant here, we've surpassed evolution, there is absolutely no reason why someone shouldn't change their gender if they want to. Apart from that, I find it hilarious how confident conservatives are about this, the situation isn't that simple, there are a number of naturally present aneuploidies including the sex chromosomes. Even in the usual XX and XY variants, parts of the Y chromosome are homologous to X and the expression of sex-specific genes is modulated by epigenetic factors. In practice the concentrations of the sex hormones are not on/off and can vary fluidly. With these simplifications, you are just displaying your ignorance of biology.