Being a hermaphrodite would mean I have both male and female biological sexual characteristics. Bigendered means my gender, not my physical sex, is both male and female.
"Gender" is a reflection of the structure of your brain. Men typically have "male brains" and women have "female brains". There's a clear physiological difference between the two (female brains have around 20% less mass, more gray matter, less white matter, and a slightly different overall structure), and that difference develops in the womb and can be easily determined in brain scans of infants, long before any social influence could manifest. Now, there are certain androgenetic mutations that can cause a male fetus to develop a female brain or vice versa, the most common of which only affects the impact of testosterone on brain development (the body already starts producing testosterone very early during pregnancy) and has no effect on muscle development, facial hair or anything else. Other mutations change the hormone balance altogether and can have more visible effects. People who are transsexual typically show those mutations, so that's a real thing, something a doctor could tell you just by looking at your brain or DNA. Transsexuals aren't special snowflakes, and if they are, it's not because they're trans.
With that said, there are indeed people who are both male and female, on a genetic level. Many of the mutations causing that lead to serious health issues, but the most common one, XXY trisomy, usually doesn't. And I would assume this mutation also affects brain development. So maybe you're that. Or maybe something else caused your brain to develop into some sort of "in-between" state. But no matter the cause, if you really perceive yourself that way, it's ultimately still physical. Free will is a myth, we're all slaves to our brains. The wiring of our brains determines how we perceive and process information, see the world, and ultimately, ourself. "Gender is a social construct", taken literally, is flat earth level mock science in my humble opinion. There are interesting discussions to be had about why boys are supposed to like blue and girls are supposed to like pink, that might be (probably is) a social construct, but this is not that.
