There is a humorous moment on a early South Park episode that points this out this odd double standard. Characters played by Cheech and Chong are believed to be "Native American" by most of the townsfolk, so the primarily white town ends up accepting and taking a great deal of silly advice from them. Eventually the characters tell everyone they are mexican, to which the townsfolk react with a great deal of horror.
Mexicans are generally a very mixed people, and much of that is from the native people of the Americas, whether its Maya or Hopi. It's why I have always hated the term "invaders" that people use to describe immigrants, it really digs into my skin more than any other term.
Oh I know they are mixed but the extent varies but there are also many that live in Mexico that may have never mixed with the Spaniards or other Europeans. My mother has an unusual last name and I looked it up, it seems to be Italian but no one could tell me how the ancestry that led to that.
And I know our past in the US may have not led to a lot if mixing with the native Americans but I also don't assume. I was raised by my mother but my father, who was your average white guy had a grandmother who was from an Iroquois tribe and lived on a reservation.
The SP episode doesn't surprise me because my mom had the same issue when my father was in the military and they lived in Georgia. It was easier for her to say she was a native American vs Mexican.
And I agree with lycophyte that people can use the word illegal or illegal alien or illegal immigrant in a racist way. Living where I live now, I'm somewhat removed from some of the racist tension in the SW US. It does sadden me. I know economic times are tough but blaming hard workers who are often exploited isn't the answer.