Misogynoir" refers to the specific forms of misogyny that Black women experience, which couple racism and sexism together. To better understand the online manifestations of this type of hate, and to propose methods that can automatically identify it, in this paper, we conduct a study on 4 cases of Black women in Tech reporting experiences of misogynoir on the Twitter platform. We follow the reactions to these cases (both supportive and non-supportive responses), and categorise them within a model of misogynoir that highlights experiences of Tone Policing, White Centring, Racial Gaslighting and Defensiveness. As an intersectional form of abusive or hateful speech, we investigate the possibilities and challenges to detect online instances of misogynoir in an automated way. We then conduct a closer qualitative analysis on messages of support and non-support to look at some of these categories in more detail. The purpose of this investigation is to understand responses to misogynoir online, including doubling down on misogynoir, engaging in performative allyship, and showing solidarity with Black women in tech.