The number I am thinking of is 80808. This is a palindrome. However, it is more than a palindrome, because it can be flipped or mirrored and will still read the same way.
Is there a word for this?
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There's a term for words written in such a form that they can be read from another viewpoint, direction or orientation. They are called ambigrams. See the relevant Wikipedia article.
A strobogrammatic number is a number that, given a base and given a set of glyphs, appears the same whether viewed normally or upside down by rotation of 180 degrees. (Wikipedia)
I think the word you may want is dihedral. The only numbers that can appear in such a construction are 0, 1, 2, 5, and 8, and the property primarily applies to numbers displayed on a calculator or using LED's.
There are also dihedral letters, namely the capitals H, I, O, and X.
I believe the term you are looking for is:
More on Wikipedia...