What is the manga about a twin brother and sister called?
I'm quite sure it is Hanayashiki no Juunintachi.

Synopsis from mangaupdates:


  Sakura Aki is a penniless high schooler who likes to cook and does other "unmanly" things. His grandfather and father are disciplinary on gender roles and beat up Aki everytime he does a thing that women would usually do, like cleaning. When his father leaves him absolutely broke, and with his grandfather realizing Aki's "unmanly" flaws, he sends him to live at one of the dorms at an all girls' school that he runs, a school where distinguished young ladies of the country attend, the Hanayashiki - Flower Estate-, to teach him a lesson.


Sakura Aki is the name of the boy who went to the same school as the two girls - the twin sister (Ayame) and her best friend (Renge).



The background story about the twin brother (Kakitsubata) and sister (Ayame) is told in chapter 8 and 9 from the point of view of the twin sister. Actually, it's mostly about how Ayame developed androphobia and personality disorder due to unhappy family, and also how she and Renge became close friends. Your description in the question matches the content of these 2 chapters quite well, so I will just put up some images to show the similarities.

This page from chapter 8 shows Ayame in the past with shoulder-length hair, and a picture of the blue bird, which their father was obsessed with. Their father also kept a bird of the same species in a cage in his room.



Another page from chapter 9 shows Ayame cutting her hair off to "solve the problem", since her hair was grabbed by her father during the previous incident where she was about to be punched. She later shaved her hair clean and wore a wig to class instead.