Interesting ... I just had an extended Facebook argument about the existence of "reverse racism". I assert that racism is NOT limited to prejudice against black people. Thus, I think so-called "reverse racism" [prejudice against whites] surely exists. My definition of racism is prejudging people on the basis of race (race is a fictional concept not recognized by modern science - see here). A strict reading of my definition precludes any meaning for the phrase reverse racism - there is only racism, regardless of the races involved.
Whites have been in the majority in the USA for a long time, and that has allowed the development of an "institutionalized" prejudice against blacks now called "white privilege". White privilege is rooted in racism, therefore, which is in turn rooted in instinctual tribalism. I can understand the reasons that might lead to some blacks becoming deeply prejudiced against whites. Unfortunately, that is basically sanctioning what I call racism. What we need to eliminate ultimately is prejudice based on human instincts embedded in our genes - it will not be easy. But it does no good to NOT have conversations with others who may have different viewpoints. Understanding someone requires an effort to see things from their perspective. Not making that effort only perpetuates prejudice.
This single phrase "reverse racism" can have multiple definitions. Your opinions about the phrase depend heavily on what meaning you assign to it. There may be others, in addition to the three I've offered below.
1. a negative pre-judgment by blacks against all whites - black against white prejudice being the "reverse" of white against black prejudice
There clearly are those who fall under #1. It can't be denied that such people exist and there may or may not be valid reasons for it. Returning tit for tat is quite understandable if you've experienced race-based injustice. However, this clearly simple racism, if you define it as I've done.
2. institutionalized favoritism for blacks
This might include such things as so-called "affirmative action" programs, which seem to anger many whites, especially conservatives. At the superficial level, this sort of thing can be characterized as reverse racism. However, the motivation for it is to be to address a long, continuing period of discrimination against blacks as a result of white privilege. It's not really so much of an attack on whites as it is an attack on white privilege. Black people deserve the opportunity to prove themselves to be competent, and if they're given some benefit of the doubt, then perhaps this is the start down a path to eventual elimination of white privilege, whereby all are always given strictly equal opportunity. It's a small price to pay for centuries of discrimination against blacks and serves many positive ends. The whole "competency" argument often thrown up against affirmative action falls apart when you realize that many white people who have been given the benefit of white privilege have proven to be incompetent! Whiteness doesn't equate with competence, just as blackness doesn't equate to incompetence. The examples (counterexamples to racial stereotypes) are all around us!!
3. opposition to institutionalized white privilege
The idea that someone opposed to white privilege is automatically exhibiting "reverse racism" is obviously fallacious. Yes, blacks prejudiced against all whites (i.e., black against white racists) certainly would be likely supporters of doing away with white privilege. Nevertheless, that doesn't apply equally to all those fighting this battle for equal justice and opportunity. Many of those seeking an end to white privilege are not at all black. Frankly, it's a position I think should be the choice of all rational people. I came to understand that the racism I encountered as a boy was not consistent with my experiences as an adult - racial stereotypes were demonstrably false - you can't claim to know anything about a particular human being solely on the basis of race. If you must judge people, do so on the basis of what they say and, more importantly, on what they do!
I observe that racism is a form of tribalism. We evolved as creatures who depend on social interactions for our survival. Those in our "tribe" were much more important to us than those from other tribes. Other tribes represented competition for resources and survival. Other tribes had different cultures, different ideas, different religions, and in some cases, had a different physical appearance. Tribalism is deeply embedded in our genetic heritage - it was an important survival trait. To be different is to be a threat.
Any social or cultural grouping can be considered a tribe, so there can be tribes within tribes (hunters, gatherers, scientists, clergy, carpenters, plumbers, etc.). Minor differences in skin coloration, eye and nose shapes, etc. have stimulated tribalistic reactions whereby those who look slightly different are seen as an "inferior race". Culturally assigned roles may have no valid basis in abilities. This sort of thinking is in opposition to the facts as we know them from science. Science tells us that all humans evolved from our beginnings in Africa - we all contain some of that original DNA and so all of us are "black" in that sense. Race has become the cultural equivalent of the appendix - it no longer has any functionality and at times can be very harmful to us. We need to discard the refuse of tribalism/racism and strive to work together for the common good. Races are mythical - there is only one race: the human race! It will not be easy to overcome our tendency for tribalism/racism, but we need to do so as soon as possible.