[slurgfest]

It's not symmetric; you are less likely to steal your girlfriend's bras. And if you do, it probably means something different from when your girlfriend steals your boxers.

Men and women do differ even from an early age, but you are of course right that these differences do not necessarily track with cultural stereotypes. (And when they do it may be because those stereotypes are being conformed to in a strongly biologically underdetermined way)