Ambonec

Ambonec is a nonbinary gender identity in which a person identifies as both female, male and neither at the same time. Coined by transmalenaoto (nbshadow) in 2014, from Latin ambo "both" + nec "neither". Described by the one who coined it as a nonbinary "gender identity in which you identify as both male and female, yet you also identify as neither, at the same time. You can lean towards any sort of gender expression [...] if you wanted. You can use [any] pronouns if you wanted. You can identify this within bi/tri/qua/etc if you wanted. You can be demi-[ambonec], meaning there could be circumstances where it changes (mood, etc). You do not need dysphoria to identify as [ambonec]." Abbreviation: AN.

Similarly, the word "genderdox," coined by Tumblr user htt in 2014 based on the word "paradox," meaning simultaneously having genders and being genderless.

An anonymous submitter to the Mogai-Archive blog in 2014 also proposed a similar paradoxical gender: oppogender, meaning "when you feel/are two (or more) contradicting or generally ‘opposite’ genders at the same time, such as being simultaneously completely gender-null and another gender. ... alternative names: contrarigender, divergender, discrepantgender."

History
In a 2004 issue of the feminist magazine Off Our Backs, a transgender lesbian named Sam Bullington wrote: "I understand myself to be simultaneously and serially both man and woman, as well as neither."