The trap sits in carta. Spanish splits what English calls a letter into two words: carta is the letter you write and send, letra is the letter of the alphabet — and in a restaurant la carta is the menu. So carta de autorización is a written letter, never a character on a page.
In English both word orders exist: authorization letter is the compact form used on websites and forms, letter of authorization is the fuller one used in legal writing, and offices abbreviate it to LOA. Spelling follows the region — authorization in the United States, authorisation in Britain. And this is not a power of attorney: when a Spanish office wants a notarised document it asks for a poder, and a plain signed letter will be turned away.