Sadly, we don't get to see who "Mannix" was sponsored by.

Here's what you would see at the opening of every episode of Mannix in the first season. The great theme by Lalo Schiffrin was a rare jazz-in-waltz-time (or 3/4 time). A waltz doesn't lend itself very well to jazz arrangement but Schiffrin was able to give it this kind of carnival ride feel to underscore how crazy Joe Mannix's life can be.

Or at least that's what my "TV Theme Analysis For Dummies" told me to say.