I'd like to point out the falsity that animals do not have gender. Perhaps crickets and pigeons do not have enough complexity within their psychology to either 'feel like a male' being a female or 'behave in stereotipically male ways' being a female (which I understand as two ways of being cross-gendered. I'm not sure this is how the term "cross-gendered" is used, but it is what I'm meaning here, having sex A and gender B)

But I'll bet all my money in that a lot of more complex animals (I"ll go with Lions, Bonobos, Dolphins and maybe Baboons) are obviously possibly cross-gendered as a personality trait. That would mean that behaviors usually pertaining to males activate in females (especially triggered ones) with strong stimuli for instance. And some specific animals (say Joe and Mimi) might be so prone to that that actually they behave more like the opposite sex than their own.

Other than that I'm happy with the above clarifying discussion.