I originally wrote the bulk of this message for the 'Bra' thread but much of it is relevent to the 'Hakama' thread as well.
The X-over top is called a kieko hanten, there are many forms of hanten, fishermen even wear a string one thats prevents water logging! it's not nessearily underwear, more a casual jacket.
Whe white pants are called Zubon, they are not underwear either, more peasants or children's play clothes. Here a century on from Kano's inintial teaching of Judo we forget that he expected the adults to wear hakama, zubon came later, there is some influence from the Shotokan and his friend Funikoshi on this.
Traditionaly under any clothing mawashi is worn literally a white cotton loin cloth. In Japan die-hards still wear these, even business men I'm told still go in for them. Yes it's an image!
As to women's clothing, the earliest photo's of Ueshiba's, Funokoshi's and Kano's female Japanese students show them in printed dress Kimono of the informal type. Usually of 2 or three layers. If we look at prints from the Mieji and Yedo periods showing women in training they are usualy going as a girlfriend of mine used to call it 'free range'. But thier training wasn't mixed.
Women wearing the same clothing as men in training i.e; Hanten, obi, and hakama or zubon came after WW2, it reflects more uni-sex training taking place, as wel as a desire for uniformity with male classes at the kodokan, shobukan and Shodokan as well as others. As well as a change in social attitudes.
Curiously untill the occupation public baths were unisex, apparently the these dissapeared in the 60's. I think there are only three traditional uni-sex baths in the whole of Japan today.
As a note many large Budo organizations stipulate in thier regulations that women should wear thier hakama longer and higher with the rim just under the bust line, and a (pretty) bow knot instead of a square knot.
The Hakama worn for training is called Joba Hakama, lit.'horse pants' they are probably a mutation of equestian Jodpers and some chinese folk dress. Thier apearnace in Japanese art ruffly matches the importation of Chinese culture during the Teisho reforms about 650ce. this matches with the social changes going on as the result of this urbanization.
I'm a Zen monk, I often wear hakama in place of my koromo when teaching informally and people still 'get' that i'm a monk, so thier is a spiritual dimension to hakama that was probably not lost on the samurai, but also they were probably valued visualy. Traditional Japanese Fashion favours a cucumber look wich is lost when wearing zubon.
basiclly you could go on making connections on fashion, history, spiritual, and functional premises forever. Does it matter?