There are two separate questions on the Census forms. The Census Bureau considers race to be separate from ethnicity. So, the first question (ethnicity) is “Are you Hispanic or Latino?”, yes or no. The second question is “What is your race?” and the options are White, Black, Asian, American Indian and Alaska Native, Native Hawaiian or Pacific Islander, Some Other Race, and Two or More Races. (2000 was the first year to offer the “two or more races” option, BTW). This means that “Hispanic” is not a race. See the official Census forms here.
So, given all that, a person can be identified by both a race and an ethnicity. Therefore, you can be Hispanic and White, Hispanic and Black, Hispanic and Asian (which is possible given that the Philippines were once a Spanish colony, so Filipinos could be considered Asian and Hispanic), or non-Hispanic and White, non-Hispanic and Black, etc. I use the analogy of being deaf when explaining this: imagine if the question on the Census was not “Are you Hispanic or Latino?” but “Are you deaf?” You could then be classified as deaf and White, deaf and Black, etc., or non-deaf and White, non-deaf and Black, etc. In actuality, I think the majority of Hispanics in our area (Dallas) don’t consider themselves any of the stated races, because we have a high percentage in the Some Other Race category around here.
This is all Census Bureau rhetoric though. I think in the end you are what you are and classification is not important.