Ytdyklly

Why is the word Gestapo feminine? Almost all other (non-abbreviated) loanwords I can think of ending in -o have been absorbed as masculine. Is it because it is associated with policía?

In all the Romance languages, gestapo is feminine despite its ending. It is most likely that whichever language first imported it (probably either French or Italian) figured that because gestapo stands in for Geheime Staatspolizei (policía estatal secreta), the appropriate use would be to make it feminine as with the analogous words la police _(FR), la polizia _(IT) especially considering that coincidentally, polizei is, coincidentally, also feminine in German. Spanish and others, in turn, maintained that usage, although it's possible for separately and independently maintaining the feminine nature of policía/polizei.

Also, even though it ends in -o which normally pushes words to be masculine, I find that acronyms and abbreviations for organizations tend to maintain the connection to the base noun's gender moreso than more common nouns.

There are always exceptions to the rule:

A very common one would be la mano

I don't think there is a specific reason why La Gestapo has la as its article.

Searching online it seems that exceptions to the rule fall into six categories:

1 - Words that are shortened versions of other words. For example, la foto (photograph) is feminine because it's short for la fotografía.

2 - Words that end in -ista as the equivalent of the English "-ist." For example, dentista can be either masculine or feminine depending on whether the dentist referred to is a man or woman.

3 - Words whose meanings vary depending on the gender. For example, in some areas, la radio means "radio," while el radio means "radius" or "radium." Sometimes la radio is used to refer to the communications medium and el radio for a radio set.

4 - Some masculine words that come from Greek and end in -a (often -ma). Most of these words have English cognates.

5 - A few compound nouns, which are traditionally masculine, even when the noun portion comes from a feminine noun.

6 - Words that are just exceptions, such as mano and día. Usually these exceptions come from the way the words were treated in Latin.

It's funny how people don't know the answer to something and still try to come up with a well explained, polished and presumptive argument. The answer for this is simple - but you won't understand it fully if your native language has no specified gender for every word, such as it is in portuguese. The simple reason is that "La Gestapo" refers to "La Policia Gestapo", and "Policia" (police) is a feminine word.
There's really no specific rule for which words are masculine and which are feminine and that's why this issue may confuse those whose native language hasn't got this aspect.