I was in a dojo in Burlington VT where men and women not only train together on the same mat, but actually dress in the same dressing room. Disconcerting at first, but somehow not very difficult once you get used to it.
I was brought up on the lie that men and women are the same, and I believed it for many years. I was very annoyed when I found out that they are very different creatures, with different brains, different hormonal systems, different behavioral patterns, different sensibilities, and different skills. The support this myth got from many circles was unconscionable: a victory of a dream over a willingness to struggle with reality.
Not that difference justifies any sort of inequality before the law or oppressive patriarchy within society, of course.