There's no upside-down "mode" that browsers or phones support — every flipped letter you see is a distinct, ordinary character that happens to have been designed to resemble another letter rotated 180°. Understanding where those characters come from explains both why the effect works and why it occasionally doesn't.
Most lowercase flipped letters come from the International Phonetic Alphabet (IPA), the character set linguists use to represent the sounds of spoken language precisely. The IPA happens to include characters like ɐ, ǝ, and ɹ for specific vowel and consonant sounds — characters that, by coincidence of shape, look like rotated versions of a, e, and r. A flipped-text generator isn't inventing these shapes; it's borrowing existing Unicode code points that were designed for an entirely different purpose and happened to end up looking right for this one.
Capital letters are harder to fake this way, since IPA doesn't cover as many of them. Generators fall back on mathematical operator symbols and other technical blocks — the "for all" symbol (∀) doubles as an upside-down A, and the "there exists" symbol turns sideways for a rough E. These substitutions are visually close but not always exact, which is why flipped capitals sometimes look slightly different in style from the lowercase letters around them.
Because every character used is a standard, decades-old part of the Unicode specification (not a custom font or an image), any device with reasonably current font support renders it correctly the same way it renders any other letter — copy, paste, resize, and search all work normally. The only failure case is a device whose installed font is missing the glyph for a specific character, which shows as a blank box instead of failing the whole line.