"Complimentary" vs "complementary"

I got a bit mixed up just now regarding the difference between "complimentary" and "complementary". My colleagues were arguing about the correct spelling of "complimentary drink" at a nightclub event, and I proposed that it be with an "e" but apparently a free promotional gift is, in fact, spelled with an "i"... Egg on my face!

My question though is if it's always as simple as this. Obviously a glass of wine can also be complementary, if it's specially chosen to go with a meal. (I wonder if any restaurant has ever offered a "complimentary complementary glass of wine", or would that be too silly?)

And it still feels to me as if as "complementary breakfast" could quite logically be one that is included in the bill, "making up the complement" of the price charged. Am I just crazy? Has it always been unequivocally "complimentary" when a service is provided free of charge as a sweetener to a deal? I kind of prefer "complimentary" to relate specifically to compliments. A restaurant owner is not really paying you a compliment by not charging you extra for breakfast, he's just doing business...

Solution 1:

complimentary: free on the house - "With compliments from the house"
complementary: to go with something

Solution 2:

Yes, it really is that simple. Complimentary = free.

A possible mnemonic device for you: If it's compl-I-mentary, I get it for free.