complimentary grammar advice

Complement: This always has the meaning of ‘completing a set/making up a whole’. This is used when saying that a tie goes well with (complements) a shirt, for example. It is also used to describe a number of people making up a group: ‘This ship has a complement of 50.’

Compliment: This has two meanings: one is that someone is expressing nice things about you, saying that you or something about you looks nice (a compliment/to compliment someone); the other (when used as ‘complimentary’) means that something is free of charge or done as an act of courtesy.

I once got this wrong in an ad and had a client point it out (they didn't catch it either) after the ad ran. Way embarrassing, but damned if I didn't learn the lesson.