People of color do not benefit from systematic racism. Therefore, “reverse racism” does not exist. However, this statement is true for a number of reasons.
Firstly, it’s important to note that prejudice is not the same as racism. A POC can be prejudiced towards white people, or have a hatred of white people, but this is not racist. This is because racism is more than just prejudice towards someone of another race. Racism is systematic. It’s institutional. White people have white privilege, and hold societal and institutional power. People of color do not have this, therefore they don’t benefit from systematic racism like white people do. Racism is power plus privilege that causes oppression. This means that people of color can certainly be racially prejudiced, but not racist.
Some people cite Affirmative Action and organizations set up for people of color as proof of racism towards white people. Firstly, equality is not racism. Taking steps to insure diversity, and taking steps to insure minority voices are heard is not indicative of racism. When people of privilege suddenly find themselves on equal ground with everyone else, it may feel like oppression, but it isn’t.
As for private spaces closed to white people, I think of it like this: LGBTQ safe spaces exist because they are needed. Every space is a safe space for cisgender/heterosexual people, but that’s not true for LGBTQ people. This therefore creates the need for an LGBTQ space where LGBTQ people can go that is safe to discuss things like oppression, discrimination, and prejudice without cishet people talking over them. This is the same for people of color. They need spaces that they can control, where they have the majority voice and can speak openly about issues that only apply to them (i.e. Racism) without white people (the group actively oppressing them) present in that space.
Oppressed groups have the right to spaces that exclude the groups that oppress them. These spaces insure their voices can be heard, and be heard safely. This is not discrimination towards white people. White people control practically every space because they have power and privilege in society, so having a minority of spaces that exclude them is not discriminatory.