I differ from the author only on emphasis, and degree. However, these differences have important effects.
Let's start with the idea that there are male and female brains. They differ anatomically, and these differences correspond to different stereotyped behaviours, regardless of upbringing. This is most obvious in girls who have CAH, masculinised brains, and a male preference in play patterns, even though they have a normal female upbringing.
Now that we've established that idea, that it's objectively observable.. we have to tear it down. It's just the first step in Wittgenstein's ladder.
The brain is a complex structure, not a simple one. Any particular individual can have a more masculine anatomy in one area, a more feminine anatomy in others. The sexually dimorphic features have considerable overlap. Within any one area, few are unambiguously male or unambiguously female. The difference is statistical. Moreover, in the higher brain, hormonal balance plays a role in changing the brain's physical anatomy. Treat a male with female hormones, some of his brain structures will feminise. The brain is plastic in many areas, experiences and social environment causes physical changes, and much of what we call "gendered behaviour" is as the result of arbitrary socially constructed factors.
OK, understood that? Now discard it, it's the second step on the ladder.
The brain is not homogenous; different parts have different effects, and while some parts are plastic, others are not. While much "gendered behaviour" is a social construct, some is not, and is remarkably resistant to change. This is most obvious in Transsexual people, whose neuro-anatomy is female in some areas, male in others, and corresponds to neither in yet others. When the Lymbic nucleus is feminised, typically feminine emotional patterns are found, and these lead to a feminine gender identity. When the Superior Parietal Lobule (SPL) is feminised as well - as it often is - the "body map" is for female primary and secondary sex characteristics - breasts, vagina etc - and any mismatch with reality causes immense distress.
This also explains such observed phenomena as "phantom limb syndrome", and the widely variable emotional response to hysterectomy or mastectomy. Also too the variable response to any natural masculinisation of a female-at-birth body due to 5ARD or 17BHSD syndromes, which can cause a superficial "natural sex change".
What this means is that a binary model of sex and gender is merely a rough approximation. A better one is a trinary one, with male, female, and bigendered (approximately 1/3 in each category). But that still doesn't capture the multi-variance and fuzziness of the situation adequately. A small percentage won't fit either model, or indeed, any such gross over-simplification.
References (to substantiate my assertions):
Sexual differentiation of the human brain: relevance for gender identity, transsexualism and sexual orientation. Swaab Gynecol Endocrinol (2004) 19:301–312.
White matter microstructure in female to male transsexuals before cross-sex hormonal treatment. A diffusion tensor imaging study. - Rametti et al, J Psychiatr Res. 2010 Jun 8.
Prenatal hormones versus postnatal socialization by parents as determinants of male-typical toy play in girls with congenital adrenal hyperplasia” Pasterski VL, Geffner ME, Brain C, Hindmarsh P, Brook C, Hines M Child Dev 76(1):264-78 2005
Changing your sex changes your brain: influences of testosterone and estrogen on adult human brain structure by Pol et al, Europ Jnl Endocrinology, Vol 155, suppl_1, S107-S114 2006
Biased-Interaction Theory of Psychosexual Development: “How Does One Know if One is Male or Female?” M.Diamond Sex Roles (2006) 55:589–600
Gender change in 46,XY persons with 5alpha-reductase-2 deficiency and 17beta-hydroxysteroid dehydrogenase-3 deficiency. Cohen-Kettenis PT. Arch Sex Behav. 2005 Aug;34(4):399-410.
I'm including this as a post because it summarises and encapsulates my conclusions so far. It's a bit more complex than this, but the 3 steps on Wittgenstein's ladder are enough.